L&l, if you live outside the zone but SW of Battersea you must live approximately between the Epsom NDB and the Ockham VOR (radio beacons), which would explain why a lot of aircraft fly nearby. Aircraft use these beacons for navigation and they are also reporting points. From operating knowledge of the area, very few rotary winged aircraft fly as low as 1,000 ft around there. Standard operating altitude for helicopters joining the London zone is 2,000 feet along that stretch and this is monitored and controlled by ATC.
I'm sure that most pilots are flying quite legally, as you have suggested yourself, and in accordance with the "fly neighbourly" advice recommended by the BHAB. The airspace in that region hasn't changed for many, many years. Unfortunately it sounds like you bought your house in an inappropriate place with regard to aviation, in view of your dislikes and opinions.
To be quite frank, regrettably for yourself, there is little you can do, apart from accept it or move house again. However difficult that is to swallow.
I do know how you must feel. We bought a property having done the usual searches plus more besides. My family and I were faced with a similar quandary, for different reasons. A local planning issue caused us major stress for years and made our house unsaleable and at times almost untenable to live in, so we had no option but to stay and get over the issues causing our problems (in my opinion more severe than yours seem to be).