There are very few places in the UK that are completely flat, but equally very few that involve really long climbs. At 8 and 10 the kids should be capable of doing some climbing - after all every climb means a chance to bomb down the other side.
If it has to be flat consider the canals and river valleys - there are plenty of good rides to be had along the canal and river network, as well as the trails on old railway lines (for example the cuckoo Trail and the downs Link are both within reasonably easy reach of Winchester).
There's nothing to stop you basing yourself at a campsite for a few days, using that as a base for bike rides, and then moving on when you are ready.
By the way the place whoes name I couldn't remember is the Vale of Pewsey - which would combine nicely with a ride along the Kenet and Avon canal.
If it has to be flat consider the canals and river valleys - there are plenty of good rides to be had along the canal and river network, as well as the trails on old railway lines (for example the cuckoo Trail and the downs Link are both within reasonably easy reach of Winchester).
There's nothing to stop you basing yourself at a campsite for a few days, using that as a base for bike rides, and then moving on when you are ready.
By the way the place whoes name I couldn't remember is the Vale of Pewsey - which would combine nicely with a ride along the Kenet and Avon canal.