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Yup - if you don't mind riding up to Peckham, then you can cut through the "surrey canal" path (there is no canal), then Burgess park. Back roads to and past Elephant & Castle and then up to Waterloo. That gives you c. 40% of the ride off road, but on tarmac'd paths. Also Rocket Van Hire on Brandon St do a nice coffee.
Like this: https://ridewithgps.com/routes/13384414
Also, if you want to cut out Waterloo at the end you can cut through to Webber St and pop out by The Cut.
Dear commuters, me and the girlfriend are having to ride our bikes with loaded panniers from New Cross to Waterloo next Tuesday morning during rush hour. I'm wise enough to avoid E&C obviously, but would anyone suggest a better route than straight up the OKR, under the fly-over to Borough and straight on? I think I'm big and ugly enough to cope with most things but she's a little timid around lorries.