Which areas in CS will be (or have been) most affected by fully homomorphic cryptography?
Before turning to your actual question, let me mention that there aren't any practical implementations of fully homomorphic encryption. There are some implementations that are far from being practical, and it's hard to guess whether that will change in the foreseeable future.

Fully homomorphic encryption is a goal in itself, as it enables outsourcing computation resources without sacrificing privacy. As such, its applications are mainly outside computer science, in the same way that the word processor Microsoft Word doesn't have any applications inside computer science (other than writing papers; though you should do that in LaTeX).

That said, fully homomorphic encryption could be used as a building block for other cryptographic applications. Vaikuntanathan's technical survey article contains some pointers in Section 6.4: verifiably outsourcing computation; and constructing short non-interactive zero-knowledge proofs. The latter has other applications and is not an end in itself, see Groth's technical survey article.