In Richmond, about 1800ft.
In Hounslow, around 1000ft to under 200ft as they cross the perimeter of the airport.
Many of these planes are large jets, 747s and similar. You can usually see the next plane before the first has disappeared from view, and hear both. Between 270,000 and 700,000 people are affected by the noise, depending who you believe.
After renting a house directly under the flight path as a student, I tried never to do so again. Places under the flightpath are slightly cheaper, although anyone new to the area probably doesn't realise the extent of the problem until it's too late.
I didn't think about it for my employer though, and worked for several years directly on the approach path. On some days it wasn't possible to hold a conversation outside. I felt it led to constant stress. On a rare day with no noise (e.g. after the volcanic eruption in Iceland) everyone seemed so happy and relaxed — as if it were the first day of nice weather in spring after a month of rain.
The airport publish this map: http://myneighbourhood.bksv.com/lhr/ which shows that in many areas, there is noise (by some definition) for more than half of the day. (Make sure to tick landing approaches at the bottom.)