Were you to park somewhere like Kew, you could cycle pretty much all of this by hugging the south bank of the river, 95% traffic free / shared path.
You'd need to do a bit of dog-legging at Putney using quiet roads to pick up the river again at Wandsworth Park. Cross the river using Waterloo Bridge, it's then a short cycle through Covent Garden to the BM in Bloomsbury.
Surface is varied - I've only ever cycled it on 23mm tyres.