I live in SE London, behind a road which is closed every year for the marathon. Every year we go out to cheer people on, its great.
There will be no hassle getting across London with a bicycle - just wait till the mass of runners have died down a bit, and police at junctions will wave you across. Just be sensible and try not to cross the route where it is jam packed with runners.
I see what you mean about Horse Guards and the Mall though - I've never been up there at the finish, so can't say, but I'd say sticking to the west of there would be better.