That's a pretty complicated question, really. It depends on many factors, and this makes me wonder if it's really a very useful question to try and answer (although I do understand why you'd ask it!)
Let's define a traditional printing session as silver gelatin - ok?
For one thing, it depends on how extensively you wash your prints. If you print (silver gel) on RC paper, not a lot of wash is needed, which saves a lot of water, potentially. FB paper will require more water, although total volume is much less relevant than flow or changes of water. You can actually wash pretty efficiently if you use a small volume and replace it a couple of times. This goes for any kind of wash - film, RC paper, FB paper, alt. process prints...
Having said that, a carbon transfer print doesn't require all that much of a wash. There's no fixer etc. to wash out of the print and there's also no risk of silvering out etc. There is water used in other steps; e.g. the initial soak / mating bath and warm water development, and of course in making the glop/tissue. For mating and warm water development, you can get away with a fairly small volume; I generally use 1.5 liters for a single 8x10" print for all steps from the initial soak through warm water development. I generally finish development with another warm water bath at maybe 750ml, and then a single final rinse at something similar. So let's say I use 4 liters of water in making an 8x10" carbon transfer, if I limit the scope to just the print. I could very effectively wash a single 8x10" silver gelatin FB print in the same volume of water. IN practice, I probably use less, since I'd rarely make a single print at a time and my water use will be more efficient if I combine several prints together in a session. This works with FB, but not with carbon. Those are one at a time.
However, I doubt that water use is really the important factor to focus on. How about energy? For making a carbon transfer print, I heat my glop to somewhere around 50C and keep it there for a few hours. I probably use up to a few kWh for that. I never really measured it, but it'll depend also on season (temperature), how long I wait for the glop to offgas etc. Then, for warm-water bath development, I boil about 800ml of water for an 8x10" print using an electric boiler. You could fairly precisely estimate how much energy that takes. We do get some of our electricity from solar panels, but how do I fold that into the calculation...? And I run a dehumidifier when drying my tissues overnight, which also eats up a decent amount of power.
With FB and RC paper, I don't have most of these energy costs - but this is only because they're expended by the manufacturer. How efficient are they? Surely, a lot more efficient than I am. But what exactly is their environmental impact?
And when I've figured all that out, how does water use relate to other kinds of impact of the printing process? Do we include waste management and potential silver recovery (or discharge into a sewer system!?) as well?
Before you know it, you're in the process of making an LCA for a print. This can be very interesting, and sure enough, this kind of work is being done in the printing industry. But to do it for home-made alt. process prints would (1) never yield a clear-cut answer because working methods and a host of other factors vary so much, (2) you'll have to make a host of assumptions and those assumptions can turn out to entirely tilt the outcome of the analysis in the end, and don't forget that (3) you lose a lot of perfectly good printing time figuring out how to weigh the different factors!
So yes, I do understand your question about water use, but I feel I have to raise some considerations relating to the feasibility of answering it, and the relevance of whatever answer we might arrive it. It's a simple answer, but underneath hides an extremely complex set of issues.
Instead, I'd propose the following: be mindful of what you do, and do the best job you can in making what you feel are the right decisions. There's not a whole lot more you can, or IMO should do.
The print with the least impact is the print that isn't made. Sadly, it's not much to look at!