Disregarding actually acquiring material and any time/effort required for the manufacturing process, would titanium cutting weapons (e.g. swords and daggers) be more effective than steel weaponry?
Let's get one thing out of the way first: 'steel' isn't a single substance - it's a term for 'an alloy of mostly iron, with some carbon and maybe some other stuff as well'. There are a lot of types of steel, with a fairly large range of properties.
Titanium could be a single substance (it's an element, not automatically an alloy), but allowing alloys for one side of the comparison but not the other seems a bit unfair. If we are allowing titanium alloys, then there's also a range of properties available here as well.
So the accurate answer is it depends on which types of steel and titanium you're talking about. If you want titanium weapons, you could definitely set up a situation where the available titanium alloys beat the available steels, but you could equally do the opposite as well.
Still, we can also do a comparison of some kind of 'average' for both sides, so let's talk about that.
Titanium generally has similar performance to steel at a lower weight
The common perception of titanium is that it's really strong, but it's more accurate to say that it's really strong for its weight.
If you have a piece of titanium and a piece of steel the same size, the titanium will be about half the weight.
Comparing strength depends on which alloys we're talking about, but it's likely that a good steel will be a bit stronger, for some definition of stronger. NixonCranium mentions that titanium is more brittle than steel, but it's more complicated than that - with 100% pure metallic titanium the problem is actually going to be that it's too flexible rather than too brittle. Getting 100% pure titanium is very expensive, though, and even a tiny amount of oxygen in the mix causes it to be brittle compared to steel. If you can produce pure enough metal, though, it forms a protective titanium oxide skin in much the same way aluminium does - it won't be automatically ruined by an oxygen atmosphere once it is finished.
Overall, that means it's likely that the titanium will fail in some way before the steel does, whether that means breaking or bending.
So what does that mean for titanium weapons?
Titanium blunt weapons are trash
I know the question asks about edged weapons, but let's cover this one as well. For maces, warhammers or similar, titanium is just too light. These weapons deal damage through their weight, so a light metal is bad. Similar logic goes for axes, although to a lesser extent.
However, if you want the classic fantasy dwarven axe with a massive blade, titanium suddenly looks better - those are actually unrealistically heavy when made from steel so the titanium could make them more plausible.
Edged weapons need some form of titanium alloy
Pure metallic titanium isn't very good at holding an edge, so hardening is necessary (and the hardening is going to make it more brittle). This doesn't have to be the full weapon - hardening the edge and leaving the body flexible is a well respected tradition in weapon-smithing - but something needs to be done.
Titanium is probably a poor choice for the army, but could potentially be great for civilian weapons.
A second effect of the low weight is going to be poor anti-armour performance, especially against plate armour. If you want to kill someone in plate, you basically have to use brute force - and a lighter weapon is worse here. This is why the age of plate mail is also the age of two-handed weapons and maces - you need the weight or your strikes just bounce off harmlessly.
None of that is what a titanium weapon would be good at - light is for fast, which matters most in an unarmoured fight where the first strike that lands probably ends it.
The people who want titanium weapons are skilled rapier duellists and knife-wielding assassins. Neither of these expect to face armour, and neither are trying to use their weapons for a full-force block against heavy attacks (even if we do assume titanium is brittle/weak compared to steel, it's definitely strong enough for a rapier-on-rapier parry). What they are trying to do is shank the other guy as fast as possible, so lighter is better.
Bonus: titanium could make really good armour
Again, not directly part of the question but we'll cover it anyway: light and strong is exactly what you want in armour.
It's not necessarily the best choice for heavy armour (the fantasy adamantine equivalent would probably still be a really high-end steel) but it should do well in the medium armour niche where you want decent protection but also still need good mobility as well (basically, it's a mithril equivalent).
No, titanium bladed weapons would not be superior to steel bladed weapons of any decent quality.
While titanium is appealing on its face due to the fact that it does not rust and is non magnetic (hence some submarine hulls are made out of it), and it’s greater hardness, there is a fundamentally fatal flaw; titanium is more brittle than steel. Being more brittle than steel is an unacceptable weakness for melee combat against enemies who have steel weapons and armor. You do not want your sword to snap in battle. The only reason you’d want titanium is if you plan on using it in saltwater or in an environment where it’s critical that it’s non magnetic.
P.S. I have titanium knives and axes and they’re cool to have. I don’t worry about rust but I’m not getting into a fight where I have to parry against steel.