I was pleasantly surprised by how nice the route to work a cycling route planner gave me in London was. There are less nice parts at the beginning and end (as I get onto and off the cycle-friendly streets), but the majority of the route (some on Q2, some on quiet 20mph roads with no specific cycling provision) is not bad.
https://www.cyclestreets.net/ is one such planner that may have bike only shorcuts that aren't marked on other map sources (it's based on OpenStreetMap and feeds cycle specific data back I believe).
