Unicomp APL keycaps are either one piece or two piece.
If they are two piece and your buckling spring keyboard already has two piece keycaps, swapping them over is as easy as popping the tops off the existing keycaps and putting the new APL tops on.
If either set is one piece, you'd just need to carefully remove each keycap including the lower half, then replace with the new APL keycaps making sure that the spring is properly seated.
That's not as difficult as it sounds, and once you've done a couple of them it will seem very, very easy. There is no need to disassemble the keyboard at all.
Unicomp keycaps are not compatible with Cherry MX switch keyboards.
A keycap puller will probably make keycap removal somewhat easier, you can either make one yourself out of paperclips, or get something like one of these:
http://techkeys.us/collections/accessories/products/keycap-puller