Still involves getting across London, and the tubes are almost all running with delays/restricted service.

The Reading/Twickenham/Wimbledon route does sound sensible. IIRC (it's a long time since I've taken that route), there atre different levels of 'slow' train between Reading & Waterloo - I think Twickenham is one of the stations included on the faster route (ie not thr train that stops at every single station)