Each soldier would be an "elite" troop, their non-dominant hand removed and replaced with a blade as a test of loyalty to their ruler and country. As the sword is a sign of loyalty they are hard to remove and removing them beyond maintenance is a sign of shame. The blade armed soldiers can be supported by normal troops.
How well would they do in battle? What battle situations would they be most effective in?
Assume this setting is medieval.
This answer says that (with the wrist intact) they'd do no better. Actually, not only would they do no better but they'd do worse, for two reasons: reduced swordsmanship, and inability to do other useful things with their non-dominant hand, like use a shield.
Effective use of a sword does not just rely on swinging the arm. The wrist is critical for first-order refinement (aiming, feints, off-side strikes, and so on), and the fingers and thumb are useful for second-order refinement. The latter depends some on the type of gauntlets being worn, but even "mitten-style" ones that don't allow good use of the fingers still afford the thumb movement independent of the fingers. This matters.
To see this, try holding something the size and weight of a sword (a wooden dowel will do). Grip it tightly in your fist and swing; observe how it moves. Now grip it loosely and swing; observe what the "sword" is trying to do. Now adjust your grip as you swing and observe that. (I'm not describing good technique here; the point of this is to show that the option space is larger if you can use your fingers.)
Further, you describe this as being done to the non-dominant hand, presumably because you want the dominant hand to remain available for other things. So, either your elite troops are fighting with two swords or they're using their weapon in the lesser hand. If they've already been trained to use a sword in the dominant hand, they'll have to relearn some of what they know. It's not just about the hand; when you switch to fighting with the "off" hand you have to adjust your stance, your power, your targeting, and your defense. (And you'll still never be as good with that hand, unless you're truly ambidextrous.) Your "elite" troops will be less-effective swordsmen than those same troops using swords in their dominant hands. If the plan is for them to use weapons in one hand they'll be less skilled; if the plan is for them to use weapons in both hands (two-sword fighting) then you've taken away the shield.
And that brings me to the second point: in battle formations (where battles are fought in open fields, as was often done in the middle ages), where troops fight as a unit, the shield wall is very important. It presents better defense when being attacked from the front, and in some configurations it protects from archery fire. Your elite troops will not be able to participate in the shield wall -- or, if they do, their main contribution will be holding shields, as we've already discussed their reduced offensive abilities.
There's more to battles than units formed up in shield walls, true -- your elite troops might be used as two-weapon harriers, who operate independently and try to disrupt enemy lines. Their mortality rate will be high (it's easier to pick off one guy acting alone than someone who can't easily be flanked or attacked from behind), but hey, they're supposed to be super-loyal, right? But they'll also be at a distinct disadvantage if they have to, say, crawl through underbrush, climb trees to drop on enemies from above, or do anything else where having both hands would help them get into position.
Also, as pointed out in this answer and in comments, the consequences of a broken or stuck weapon are much, much more severe in this scenario, as are the consequences of injury to the dominant hand.
And on top of all that, as others have noted, you're impeding their ability to do the tasks of daily living (like getting dressed) and ruining them for non-military tasks completely -- maybe ok if you're planning for them all to die in battle, but not very versatile.
Those so-called elite troops would need normals to pitch a tent, climb a wall, drink from a canteen, or tie their shoelaces. Something like that might work for a gladiator, but not for troops in the field.
BAD idea.
First, without antibiotics, the operation to create such a soldier would most likely kill half of your force (if you are lucky).
Second, your soldiers would have an extra knife/short sword but would be unable of using heavy/long two handed weapons (like combat hammers, polearms, bows).
Third, as @o.m. points, they would be unable to perform most of the operations of the camp.
Fourth, even the slightest injury to their unmodified hand makes them almost unable to attend even themselves ("soldier dies wiping his ass with accessory hand").
Fifth, in the event of battle, if the weapon breaks/becomes dull from usage, they have basically lost an arm.
Sixth, your soldiers may object to being impeded to even going back to civilian life (even if they get too old or another injure forbids them from going to the battlefield).
Seventh, you have to pay sustenance to those troops even in times of peace. You can't just send them away to grow some food.
And eighth, getting away with the wrist means that the path of attack of this weapon is very easy to predict (just follow the line of the arm) and block.
I don't see why they would do any better than someone holding a sword, perhaps strapped to a gauntlet or glove.
If the wrist is missing or compromised, I would think a lot worse.
Even with a good wrist, being able to twirl the sword within the hand is a degree of motion beyond that of the limbs alone.
I believe the other answers have sufficiently identified the limits of this approach. In the interests of having a "yes you can do it" answer on the books, I would like to point towards the works of Frank Herbert. I am unaware of any soldier intentionally mutilating themselves in this way, but he did explore the psychological side of combat (and politics... though I admit it is hard to tell the two apart in his style of writing). There may be some psychological ploys to be had.
Consider a less than perfect opponent: one that has mental quirks that still need to be ironed out. If you can take advantage of those quirks, you can find their weakness and destroy them. With enough political, religious, or otherwise mythological work, you could instil a terror in the opponents so deep that it might outweigh the severe combat penalties of your barbaric ritual. Consider a myth that, due to the lack of a hand to hold a shield, these elite solders are incapable of simply holding a line... so they must always be in the process of killing you. You have an army that will never retreat, only advance, because that is what the blades demand.
This would have a chance of unbalancing opponents, for they feel they are against an unbalanced foe of brutal capacity. When facing such a foe, you tend to unbalance yourself, and that's where the trap would have to be laid. You would have to train these soldiers to remain mentally balanced at all times, even while staring death in the face, sans a left hand. If they could maintain this illusion, they could build up the myth of their own superiority. The tiger in pursuit remains in control, the deer in panic flees wildly.
I say this is a Herbert style intrigue because the sword cuts both ways. The trainers and the king know full well that they haven't made their elite soldiers better. They have intentionally maimed them. As long as that secret is safe, the swordsmen are immortal on the field. If the regent were to let that information slip, the swordsmen would quickly lose their illusion and be destroyed. Thus there would be great political intrigue to be had. The soldiers need not know their own weakness, but at some point the trainers would need to know. Thus there would be an entire class of individuals who share a secret which could tear apart the most feared legion in the land.
Hands are one of the most versatile and important part of any next to their feet to carry them to battle. There are so many things that the hands and fingers are capable of, that this seems much more like a punishment to a soldier than a mark of honor. In general disabling a soldier especially an elite soldier is a poor idea. You could get the same kind of 'honor' status by having special weapons that are affixed to their arms that have a spring loaded blade, in medieval times a good high quality blade that is spring loaded and can be retracted would be a very valuable and prestigious weapon to be in possession of. That would be a great honor and wouldn't require mutilation.
Of course if the person awarding these honors is a terrible monster of a person then, suffering pain for loyalty and losing many to infections might be just what you are looking for.
Aside and in addition to the previous answers...
(1) There are many duties a soldier has to do that are indirectly related to combat. From dressing themselves to eating food. Not that these things are impossible to do single-handedly, but you might want to have assistants (like medieval knaves, but for everyday tasks) in your story.
(2) Battle involves more than just sword-fighting. Although there is a romantic vision of the knights in an open field, there were most certainly times when the majority of fighters (non-knights) required climbing and crawling skills. You'll need to ensure that these fighters stay as elite as knights in your story, largely ceremonial and/or for specific battle tasks.
(3) There are many un-battle-related tasks like sleeping and sex whereby it would be impractical to have a sword in the room. This isn't your question, but it should be considered: maybe they can remove their swords more often than you suggest?
Bad idea.
Among all close combat weapons, swords require the most agility. Despite videogames conditioning us that swords require strength and bows require agility, actually it's reversed. Despite movies and videogames showing us fighters who hack at each other sloooowly and with full swings just as if they were chopping wood with an axe, swords were never used for chopping. They were used either for thrusting or cutting, both requiring great wrist agility.
See it for yourself in this video. Think about how you could perform those quick motions with a sword rigidly fixed to your arm, without your hands.
Against armor, you would be in even deeper trouble. Using sword against armor is possible, but it requires special tactics which would be impossible without being flexible with your hands. Even just basic stabbing would be inefficient, you would not be able to pull your sword back far enough, so an opponent stepping in close would make you completely defenseless.
Besides these disadvantages, medieval warriors needed to be proficient in many types of weapons, as all weapon types have advantages and disadvantages. Swords, maces, pikes; open field, ambush, siege or indoors, countering infantry or cavalry, different situations require different weapons and tactics.
I will go against flow little bit, and I say, during combat it actually might be useful. In fact, excluding that it is permanently mounted (which that sounds really terrible, as that kind of soldier is useless outside battlefield), it looks similar to Pata sword
Others have already shown that this is a bad idea if you want them to be effective in combat. However, there are situations in which such a mutilation may aid in a battle: Psychological warfare.
If you're on the front lines and you've heard rumors that the enemy whips bezerkers into frenzy and then cuts off their freaking hands so they can strap on swords (I recommend both hands for this), you're going to be terrified. Remember that medieval armies are made of people and people can be made to be afraid. If you can break their morale, you can scatter them and then do some real damage on the retreat.
Imagine a battle scene where the psycho-army is already winning a bit. The enemy is already tired and on the defense. The front line parts a bit to allow the lathered, screaming, sword-arm lunatics through. The opposite line is not going to stop to ponder the ineffectiveness of such an offense; they just want to get away from the crazy person.
There are two ways you go with it: 1) Lunatics are really sworded up and are either killed after battle or simply pursue the enemy until killed OR 2) Lunatics have sworded gauntlets strapped to their arms which are removed once they've calmed down from the bloodlust. Of course, with option (2), you'd have to be very sure to hide the secret from your enemy.
Predators has unremovable wrist blades too. But they do not have amputated fingers for it. So, i recommend you to make your elite troops having something similar with intact palms of hands and watch Predator movies to see how do they operate their wristblades)
The audience would care too much about having perfectly healthy soldiers have one of their limbs amputated.
This idea could work if the soldiers were:
Otherwise, having a sword for a hand would be a detriment to the soldiers as pointed out by the previous answers.
