I think NCN-4 is the best bet, then head south from Vauxhall.
The trouble is that most cycling organisations set out to be idealogical/leftist/democratic and then they all seem to get a bit little hitler about their own domain and not want to share it.
Hampton court, yay green spaces, the environment, etc etc. But the NCN route has to officially go on a big detour around it. Just put it on the bloody map, with a note to say you weren’t allowed to plaster every lampost with a little blue sticker!
Then at the other end of the scale there’s sections near my house that are BW’s and so overgrown they’re impassable during the summer, and a mud bath in winter, but someone in an office somewhere saw a BW and stuck their purple highlighter over it (and probably got a bit of a grant/funding as a result).
In fact most of the sustrans map is pretty useless. Just give me a bloody map of paths and quiet roads that meet some arbitrary definition like passable by a nervous inexperienced cyclists who can’t carry their bike over a stile, or boardmans 8 and 80yr old test. It would probably be cheaper than sending out the crusty hippie to stand on my commute once a month to try and get me to join your organisation (I declined when the main benefit seemed to be getting my own paper copy of the aforementioned useless map).