Not that many at Greenham itself but north west of Newbury (a couple of miles away) there are loads IIRC. I've not done them for years but I remember spending alot of time around there, also your only 30mins from Salisbury Plain plenty there too but that is really Bakerman territory
You could PM Clive he has ridden and listed on here alot of the legal lanes around Berkshire, Hampshire and Oxfordshire
Its all pretty local to me so if you need owt give us a shout