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Putney to Southbank is my commute and came to the same conclusion.
Putney Bridge, New Kings Road, Embankment and I go over Lambeth Bridge, Lambeth Palace Road, York Road, Chicheley St, Belvedere Road, Upper Ground.
Coming back I continue along Belvedere Rd round the back of County Hall and over Westminster Bridge, Parliament Sq and Abingdon St...
For Liverpool St I'd go over Lambeth Bridge, Lambeth Road, Borough Rd (bikes can get through by the roundabout), Borough High St and then London Bridge. But I've only ever done this at 5am at the weekend when riding up to Cambridge and I can imagine Borough High St being a industrial sized bag of stab. No idea what it's like during normal commuting times so take this bit with a huge pinch of salt.
100% agree with amey.
Putney <--> Canary Wharf is my commute. I used to stay sarf to Southwark Bridge, but fuck that. Fuck Vauxhall roundabout, Waterloo roundabout and the road leading to it, Southwark high street, and especially Battersea high street. No thanks.
Putney bridge (or Wandsworth if you're east side massif) --> New King's Road --> follow river to Blackfriars. Fewer punctures, traffic lights and high street nastiness - choose a sprint along CS8.