Near Heathrow T4
Finding good walking spots within striking distance of major international airports isn't often easy. This is a modest area - a couple of joined fields with a bit of wood between them, but it's just a very few minutes' drive from Heathrow Terminal 4. Exceedingly handy if you're waiting for someone to come through immigration and customs, as I was!
On Oaks Road in Stanwell, there's a turning into Russell Drive, and just beside this is a very uninspiring gate (see photos), with a squiffy sign saying 'Heathrow Conservation Site'. The field behind it, though was greatly enjoyed by my dog after a long motorway journey.
Also, just around the corner in Oaks Road is the excellent 'Heathrow Cottages' B&B, which will sometimes allow dogs to stay. Recommended.
Update: If you want to be near Heathrow but can go a little further afield, Harmondsworth Moor is a great alternative.