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Chalfie-
I've cycled it before when working out that way it's better to go slightly north and go along Greenwich and Deptford Bridge way or even the other one Hornfair and sun in the sands(maybe bias it's my favourite roundabout) but Old Dover then Langton way and Vanbrugh Park can get you around into Greenwich Park avoiding it then go via tunnel and up to join the far east parts of CS3.
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Not a huge fan of the Lewisham ‘gyratory’ and the Peckham/New Cross area isn’t that much fun but overall it’s not bad.
Previous commutes from that side of London have been in from Royal Victoria Dock and Charlton, so may well give it a go at getting Greenwich way earlier (the CS3 way via the tunnel sounds like it’s worth exploring too).
Clockwise
tonylast
Might have to try this instead of my current Dartford > Westminster route that basically goes in a straight-ish line via Bexley, Blackfen, Eltham, Peckham, Camberwell and Vauxhall...