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I've got basically the same start and end point and have been honing my route for the last year or so to avoid busy roads and traffic lights. It's a bit longer but so, so much more pleasant and same time/quicker due to the lack of traffic and lights:
https://goo.gl/maps/qdsoTYxqktxiSRVv7You can get from Beckenham to Bricklayers Arms before you hit a traffic light, and Waterlink Way / Q1 is pretty relaxing to cycle on.
Seem to be unable to find anything via search but looking for some route advice. Currently doing Beckenham to the City via Palace>Herne Hill>E&C>Southwark etc. Moving every so slightly further East which will make braving Penge not really worth it, plus more bumps than I have energy for these days, being lazy and all.
Basically looking for a route from the bottom of Bromley Hill (A21) into the city. Can easily divert through Beckenham Place Park which I probably will do anyway.
Any suggestions - something simple preferably?