You can cycle from the Tate Modern south of the river to Clapham in south west London entirely on quiet streets and protected cycle lanes.
The route is 7.2km long (4.5 miles) and makes use of Cycleway 5 and Southwark's Great Suffolk Street LTN.
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You can see a digital map of the route and download a GPS/GPX file to use on whatever device or app you want here:
https://www.komoot.com/tour/201713920...
And you can find a viewer-created and maintained map of all the London Cycle Routes videos here:
https://www.google.com/maps/d/u/0/vie...
I also highly recommend the Safe Cycle London map for route planning, which is compiled by @SafeCycleLDN on twitter:
https://www.google.com/maps/d/viewer?...