I live nearby and work in Newbury. If you're after somewhere quiet to ride then explore Greenham Common, there's some nice singletrack (ableit no much) if you stick around the perimeter, and it's rideable from anywhere in Newbury.
Swinley's not too far, you've got Wayfarers walk to the south, South Downs Way starts at Winchester, and Micheldever/ Black Wood near Winch too. There was also a proposal for a mtb centre at Farley Mount, again near Winchester, but I'm not sure what's happening with that at the moment.
C_G is definately the lady to speak to regarding routes around the area, she gave me some brilliant advice a year or so ago 🙂