Lots of BA pilots, both short and long haul, live nowhere near London. However what you gain by being away from the South East you can lose in a long drive to and from work when you're generally likely to be knackered in one or both directions. Early starts and late finishes can be a problem as you'd either be driving at an unpleasant time or the flight schedules won't tie up and you'll waste a night sitting in a cheap B&B somewhere.
You're right about short haul being away from home a lot. BALPA did a bit of research into this last year and found that both short and long haul pilots in BA spend a broadly similar amount of time away from base with a similar number of nights away from home, however short haul actually spent about 30% more time on duty than long haul. It's also worth noting that many long haul flights depart late at night and return early in the morning so even though your tired you can still do something at home on the last day of the trip. With short haul you'll leave early morning and arrive home late at night so the day's completely filled. If you want time at home then long haul is by far the way to go.