I'm more interested in discussion of entire karate ryu or ryu-ha that adopted TKD kicks. I just don't think that happened by and large. Karate systems have considerably different tactical goals than TKD systems to. It would make little sense for most of them to borrow things like ax kicks or jumping 360 kicks.
I am a bit curious to the origin of the axe kick. We do it at higher kyu levels at our traditional Wado dojo, certainly not the most practiced kick but is is in the actual curriculum (rather than an exercice meant to improve balance and flexibility). Sensei tells us there is a lot of stuff borrowed from other styles (Aikido, FMA, Tai Chi etc), but TKD is not one of these. I wouldn`t be suprised if there are other applications for the move besides the actual kick tho.