Thanks for the replies. I examined the rear cogs on both bikes, and the Record looks nearly mint - I guess they were making them out of pretty strong stuff back then. So maybe just a chain will suffice. On the Bianchi, most of the cogs look good. One of them has one sort of funky looking tooth, but that may have just been the light.

I'm realizing that buying old bikes can turn out to be pretty expensive, if you want to try to make them perfect.