To set the circumstance: I play Akshan, mischievous and take my opponent's B.F. Sword on my turn, on their following turn they play an Akshan, mischievous and take that B.F. Sword back—if at this point I then kill their Akshan, mischievous, what happens to their B.F. Sword?

You regain control of the B.F. Sword.

Here's why: When you played your Akshan first (stealing the sword), you created a control-change effect that lasts "until I leave the board." Your opponent's Akshan created a newer control-change effect that overrode yours via timestamp order. When you kill their Akshan, their effect stops applying. The game recalculates control, and since your Akshan is still on the board, your original control-change effect becomes active again — you take the B.F. Sword back.

(FAQ #8851 and #10776 both directly confirm this result.)