That’s a nice route. I’m a very fast cyclist, so I can do Waterloo-Shoreditch in about 15 minutes. So I reckon 30 to Mile End. If you’re a newbie, allow 45 mins (once you know the route!).
The first half of that is through the City of London, so do it on a Sunday when you’ll have the streets to yourself. Then come back the same way. Do that over a few weekends, and you’ll be all set for the weekday commute.
For the quietest route, I’d suggest:
Pick up at the massive docking station at Waterloo.
Cycle along the south bank (shh, it’s technically illegal, but…ya know!)
Cross the river on the millenium bridge (WHEEL it across! – whatever you do, avoid blackfriars bridge, it’s one of the most dangerous spots in london)
Go straight ahead
Duck round the back of St Pauls
Follow the one way system (wheel it on the pavement if you’re nervous here)
Turn right at Barbican (fairly wide cycle lane through here)
Cross over Finsbury Square and turn left into wilson street/paul street – nice and quiet with cycle lanes and cycle contraflows
Cross over at the pedestrian/cycle lights and turn right into rivington street
At the end, go down calvert avenue, around the pretty arnold circus
Then along bethnal green road and roman road until you get to the canal, then it’s a quiet canalside cycle to mile end.