In reply to Dan Arkle:
Surprisingly hard to recommend. The Derwent is the obvious choice, Darley Dale down to Matlock Bath, but is quite short and access not always easy.
The Idle from Old London Road nr West Drayton, through Retford to Chainbridge Lane to the north, is a nice interesting day out but not whitewater as such. Does need equivalent skills to G1/2, though just because of a few small weirs (runnable in most conditions) and likely tree dodging/blockage hazards in moderately fast flow. In high water, you won't get under some of the bridges, and it should be avoided.
Downstream from there is really just a pleasant small rural river without features.
If the water levels aren't right for the Idle, the nearby Chesterfield Canal is lovely and rural to be honest. I'd recommend Wiseton, through Drakeholes Tunnel (short but interesting) to the top lock for the nicest bits, extendable to the south if a one-way trip.
A different option is that Ladybower res has recently opened up for paddle sports; its a pay to paddle option though.