Changing the crank should have no effect on how the rear derailleur works. Assuming that the rings are the same, you would be able to reuse the chain, but I would not. An inexpensive SRAM chain with its master link makes it simple to install (after sizing), and should give good service. If the chainrings are not the same size, you might need to replace the chain anyway, so why not just 'bite the bullet' and do it anyhow?

This all presupposes that the BB coming out and the one going back are both the 'standard' English threading, of course. If the large chainring is more than 2 teeth larger/smaller, you will need to raise/lower the front derailleur for the best shifting.