I commuted to work in central London from the NE for 2 years, and regularly cycle across between train stations.
I have no problem with riding there. The road network is quite complex, so you need to pick your routes with care. You can avoid most of the big roads, but even then, the speeds are not so high. The canal tow-paths make excellent commuting routes, as do the small networks of lanes such as Soho or the East End. Riding at night in London is a real joy. It seems to be much safer than walking or taking public transport late at night.

Some parts of London are harder to ride than others, but compared to other towns and cities in the UK, it is very manageable. You need to use your road sense, which many riders seem to lack.

I took my gf on a cycle tour of London one Sunday. She is not very confident on a bike, but had no problem along my routes with quiet Sunday traffic. I can recomend it to any tourist or visitor who really wants to see London.