I've been crossing the Atlantic to/from southern UK at my own expense regularly for a few years now, and in my experience the best fares are to be found in the 6-10 week window before the travel date. It always helps to check the "my dates are flexible" option when searching for flights. There are often some good deals to be found there. I alternate between BA and United, however, being a Star Alliance member, there are some good deals to be found on United if you don't mind a short connection via, say, Brussels or Frankfurt.