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This.
I live north of woolwich and when I was renting at South of the river, the roads seems better and travelling from woolwich to Greenwich you get a lot more people on bikes riding together. North travelling to west isn't great considering you got to navigate to towards the blackwall tunnel/popular area which isn't great at rush hour. Plus not many cyclist on the road riding together (probably a handful compared to the south of the river)
HoKe
If it was me: from Holborn Viaduct heading west, turn right onto Grays Inn Road towards Kings Cross. Avoid the last awful bit of the junction by turning left into Argyle St (bike lane allows you to go against the 1-way street) and following it to the nice crossing between the lights on Euston Rd, making turning up Pancras Rd easy.
On the way home it's usually more flowing in the other direction so I'd use the main roads for the 1-way system in front of Kings Cross, eastbound on Euston Rd onto Pentonville Rd, get in the right-hand lane for Kings Cross Rd, follow it round and south it turns into Farringdon Rd. And then to avoid Smithfields I'd cut left on Farringdon Lane (mind the step in the road, take it perpendicular!) just to avoid the crossroad, left on Clerkenwell Rd, right on Goswell Rd, past the Museum of London still heading south, then left on Gresham Rd or Cheapside to get to the Bank.
out of interest, why cross at Woolwich and not one of the central bridges? (Blackfriars, Southwark or London would all work) The Old Kent Rd is a bit slow in rush hour, but then the Lower Lea crossing, East India Dock Rd etc isn't all that fun either.