As described, zombies would not make a significant difference to most combat.
Firstly, they are not available in large enough numbers to send overwhelming hordes across no mans land. Taking the British Army as an example (for no better reason than their casualty figures came up first in Google...) the British mobilized 8.375 million men in total. Of those, something over 702,000 were killed. However, many of those bodies were unrecoverable, and still more would, presumably, not have been suitable for reanimation due to excessive damage at the time of death. Our number of zomibifiable bodies will have to be a bit of a guess, but based on the rule of thumb that about one third of a combat unit will become casualties, one third of the casualties will die, we could extend the rule of thumb to say one third of fatalities cannot be reanimated. That will leave us with around 468,000 zombie troops - throughout the entire War. If over 8 million wasn't enough to sweep resistance aside, another half a million probably isn't going to change things all that much.
Plus ALL sides in your war have this capability, therefore one side does not gain an advantage the other lacks by doing this.
Zombies simply aren't all that useful as infantry. They can't operate firearms, so all they can do is advance across no mans land were they will be mown down by the enemies artillery. Humans will at least crouch and try and use what cover they can to advance. As described it seems that the zombies will do is walk slowly towards the opposing trench line making them extremely susceptible to shell fire. I would also imaging small arms fire hitting the legs would probably immobilise a zombie in the muddy environment of many WWI battlefields.
The best I can see is that after a successful attack the attacker takes some time to regroup and can now make a second push, but this time about one in three of the advancing troops is a zombie that can at least soak up a bullet that might have hit a still living soldier. However, you have now diluted your capability for fire superiority, so even that might not be an advantage.
Then there is the morale issue - imagine the horror of seeing your best mate get killed, only to find yourself standing next to him again. Of course, you could mitigate this by only raising the enemies dead and give them the problem of shooting at their former friends.
So zombies don't appear to be much use as assault infantry. Are there other military tasks they can perform to free up other soldiers? Again, almost any task worth doing requires at least some intelligence and self direction. They might be able to assist with casualty evacuation - a stretcher bearer that doesn't mind getting shot too much is definitely useful! Anything that increases the survival rates of wounded soldiers is good (although again this is only a force multiplier if you can do it and your opponent can't).
Zombies could potentially replace horses as draught animals where speed isn't an issue, and shortage of horses was a real problem, but that is still not going to win the war.
Zombies might have use as a terror weapon - send a small unit into the enemy's rear, massacre a bunch of civilians, reanimate them and then set them loose. The enemy has to divert resources to clearing up the infestation.
Collecting bodies to turn into zombies would be a good task for zombies to do - particularly since you could never know if the pile of bodies were really dead, or were actually zombies waiting to attack who ever came along to clear them away. (Escape and Evasion is left as an exercise for the necromancer in this case...)