I do not envy you with this one! Could you put up those triangular "wet floor" signs? Plus directions to the other bathroom??? You cant possibly get a floor clean with people walking on it while wet.
I think what I would do is this and it may sound silly. If the floor is damp, theres not much point in sweeping or vacuuming. I would start mopping, and puishh the junk, hair and trash into piles which I'd pick up with paper towels. Then, after doing that I would truly mop to get it a lot cleaner. UGH! Public bathrooms!