You can cycle from Leytonstone in east London to Westminster in central London entirely on quiet streets and protected cycle lanes.
The route is 14.4km long (9 miles) and makes use of LTNs in Waltham Forest and Newham, as well as Cycleway 2, Cycleway 3, and the Mansell Street link.
The journey takes around 50 minutes to ride, which is roughly the same time it takes to get public transport between these two places.
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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/1275771500?ref=wtd&share_token=aXRCd90N1n3RVZ1P63wpi3gk775axxsjfYuaE8jefc9BNPUThf
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/viewer?mid=1h9Hxm57fPvZmcuSXajM_Wu0G0s6f_bs&ll=51.505213496092054%2C-0.1285238120117249&z=12
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?mid=1XlpvN9R-Wg7qZHyezO8y-eVlftr4e0WX&hl=en&ll=51.516975804561284%2C-0.21828576419061996&z=11