Savernake has some good singletrack, but very much needs exploration or knowledge – it’s not been developed for cycling or walking really at all (which is part of the charm). You’ll be spat out by the picnic site from the railway track which is as good a place to start as any, although it’s a bit of a slog from Swindon proper (easy enough, but a time sink).
West Woods would be worth a drive and ride, especially to head out south to the top of Martinsell to admire the view (cheeky trail warning – never had any trouble though).
You might have to link the BW you’re thinking about – I’d normally go over Fyfield Down that way rather than in the river valley, it’s probably a bog fest this time of year with the springs up and horses out.
I know of 6 horses – Uffington, Cherhill, Marlborough, Hackpen, Westbury, Alton Barnes .. google White Horse Challenge if you have the road bike.
Go see Sean @ Bertie Maffoons in Marlborough if you are here any length of time. Might still be an evening group ride into the forest going on. I’d be happy to offer but I’m 30 minutes SE of Marlborough now and work is interfering with my free time :-/