Hampton would be a convenient location, but as far as I can tell it is like a car park/housing estate as it seems to be full of young professional couples who all have two cars, but only one parking space.
As mentioned, somewhere like Yaxley, Stilton, Sawtry would be good or perhaps somewhere near Ferry Meadows; The Ortons area – just avoid Orton Goldhay/Malbourne area.
You could also look a bit further afield like Market Deeping/Deeping St James, but that will up your commute by quite a bit. But, rental accommodation is well priced around there.
As for riding, particularly on the road Jamie’s arrow map is a good guideline. Peterborough seems to be placed on this borderline of good/bad road stuff. If you go East it will invariably end in headwinds, boring straight roads and you will potentially lose the will to live.
Go west though and you have lots of villages and even the odd hill here and there. Towards Stamford/Oakham/Rutland water there’s some good riding.
Regarding off road – can’t really comment around Peterborough as I just don’t bother. Most of the bridleways round my way seem to just be dead ends.
I ride Rutland fairly regularly. It’s a good fitness loop, just make sure you get there early at the weekend and out before the families arrive and do a good job of getting in the way.
There’s Thetford about 1hr 20 mins away, Wakerly is OK but gets very boggy in wet weather and Cannock is quite a bit further afield.
I live just north of Peterborough and can get to Hope for a Peak district ride in about 2hrs 20mins.
I am also surprised there is a Peterborough mtb club on Facebook!