Actually, the meaning of "the second mouse gets the cheese" is that the FIRST mouse gets no cheese because he's
dead:
Which is to say that the cheese is understood in this context to be the
приманка ("bait") in a
мышеловка ("mousetrap")!!
Значит, мышка, которая рано встала с постели сдохнет в мышеловке; а мышка, которая проспала, украдёт сыр не в опасности.
(I mean to say "The mouse who got out of bed early will die in the mousetrap; the mouse who overslept will steal the cheese without danger." -- corrections welcome, please!)