The nearest to Bedfordshire would probably by the Peak district on the Darbyshire/Yorkshire border. Or for forests there's Thetford forest near Norwich. Unfortunately, the nearer you are to the south-east the more built up the country becomes, all the wilder remoter places are in the North or West. There are lots of small pockets of unspoiled land even in the most densely populated parts of the country, there's the RSPB reserve at Sandy just south of Bedford, for example.
Wild swimming is I'm afraid looked on as something beyond the pale in Britain, no lifeguard, untreated water, there might be currents, deep water, rats. But then lots of us still do it - quietly.