If you are anything like me, then a certain amount of time spent standing at junctions is inevitable, I spent my first week commuting doing this...
Bicycle Path - Oxford Circus to Stockwell at Bikely.comThis is my commute home, which takes in most of the route you would need. I am just a bit south of the Euston Road at work, and Brixton is just up the road from where I live...
Vauxhall could be hairy because you do not know which lane to shoot for. Doing it for the first time, I would get off at Vauxhall railway station (as the road curves round to the left and you leave Albert Embankment) and use the cycle path and toucan crossings to get you onto South Lambeth road.
They are rather annoying cycle paths, but when you don't know where you are going, probably safer.
IMO Lambeth is one of the best bridges to go over when you are trying to get to that bit of Sarf London. Quieter than Vauxhall, which I never use going southbound any more.