Max, Hi, I'm an ex-Cov Atco with 2 bits of (possibly)useful advice
1 Avoid areas like Binley/Willenhall/Stoneleigh like the plague, 748's and DC6's are very unpleasant at 3am.
2 Leamington can be quite expensive as can Warwick but Cov is OK, see if you can get somewhere in and around Finham/Styvechale, they're only 2 miles from the airport but get very little nighttime noise. If you are having trouble, ask Kerry at ATA if she knows of anywhere (that's if she still works there)! The ATC unit is quite good as units go, they're a pretty decent bunch. Good luck.
Sleeping's permitted, dribbling isn't!