New Forest is ideal for this – bike hire, easy tracks, easy to get to by train, loads of nice places to stay.
Or, Ashbourne in the Peak District and ride the Tissington Trail (easy ex-railway track with bike hire), although you need to get train to Derby + bus to get there, so you'd probably want to stay over to do that.
Cannock has easy family trails as well as follow the dog, although it is a bike ride away from the nearest station.
Swinley is also a bike ride away from Martin's Heron. It'd be a pain to get to though, probably need to change in London or somewhere, take you a couple of hours or so.
Maybe your best bet is to borrow a bike off someone local so you can ride round there, where you know where you're going, and it is nice and flat. Or even buy a cheap 2nd hand bike – if you add up the cost of train tickets etc. to go to one of these faraway places, plus the cost of bike hire etc. I bet it'd be way more than the £50 or so you'd need to get a 2nd hand bike capable of going on the local canal towpaths?
Joe