Often flights out of Heathrow and Gatwick can be cheaper than going out of somewhere like Bristol or Cardiff. Even though I live in Bristol (or at least I did until this week), I usually fly to the US from Gatwick or Heathrow, because of the price differences.
Doing a quick search on Kayak, the cheapest flight from Gatwick on those dates is £438 per person (1 connection), but the cheapest flight from Cardiff is £541 per person (with 2 connections, one in Amsterdam)!
Manchester has flights starting at £463, and the cheapest from Birmingham is £476 (two connections, one in Frankfurt).