Saying "F1= (.7373) base n" means F1= 7/n+ 3/n2+ 7/n3+ 3/n4. Similarly, saying "F1= (.5252) base m" means that F1= 5/m+ 2/m2+ 5/m3+ 2/m4. Putting those together you know that 7/n+ 3/n2+ 7/n3+ 3/n4= 5/m+ 2/m2+ 5/m3+ 2/m4. Likewise, F2= (.3737) base n and F2= (.2525) base m gives 3/n+ 7/n2+ 3/n3+ 7/n4= 2/m+ 5/m2+ 2/m3+ 5/m4. That gives you two equations to solve for the two unknown numbers m and n.
Edited to replace "y/n3" by "7/n3". My finger slipped!
It seems to me that you're supposed to spot the weird symmetry in the numbers here. '7373', '3737', '5252', '2525'. HallsofIvy outlines a general method of solving this problem when the given numbers have no special pattern, but f:grumpy:k only knows how you'd solve those simultaneous equations!
My thoughts. Try adding F1 and F2 in base m and in base n. Assume, just to make it easier, that n>10 and m>7 (if that's not the case I'm sure it'll turn up in some contradiction or other)
(eta: well, I'm not making much progress... but I still think that it's important to note the symmetry here. It means something, I'm sure of it.)