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Here's what google recommends from Kennington to Lyons.
https://www.google.co.uk/maps/dir/Kennington,+London,+UK/Lyon,+France/@48.5387096,-1.8278208,6z/am=t/data=!3m1!4b1!4m14!4m13!1m5!1m1!1s0x4876049103b39bb1:0x27a6cd830f527961!2m2!1d-0.1115584!2d51.4879638!1m5!1m1!1s0x47f4ea516ae88797:0x408ab2ae4bb21f0!2m2!1d4.835659!2d45.764043!3e1 -
When Tower Bridge is open I go up to Elephant, NKR, Tower Bridge Rd, Bridge, CS3, left up to Cannon St / New Rd, Whitechapel, Cambridge Heath Rd, right at York Hall, left onto Approach rd, along the top of Victoria Park, Wick Way, Eastway (bit hairy joining the multi carriageway by the Hockey Stadium), then I jump off behind the bus stop after the lights at the Leyton sign and weave round the back near Orient footy stadium...
So essentially I'm on the main roads too, apart from the park bit...
Yeah I try to avoid but Kennington to Leyton when Vicky Park and Tower Bridge are closed generally narrows the options somewhat.