I lived in London for a year just recently. During this time I commuted by bike and usually used it as my main means of transport. Unless you're going very long distances it is far quicker than tube/bus and walking. You avoid jam packed tubes in the morning and the centre on Saturdays. It also means that at night you can cycle home rather than having to walk through dodgy areas to the nearest tube station. I even managed this without incident whilst drunk an undisclosed number of occasions, but that is something I would reccomend against doing. There are so many sets of traffic lights and general jams that the traffic doesn't often pick up too much speed. There are loads of bus lanes which cyclists are allowed to use. Drivers can be arsey and tell you to get off the road etc. etc. -just ignore them. My advice is to cycle very positively/firmly, make it obvious which lane you are in, where you are going, and to get there quickly. Nothing winds a motorist up like a indecisive cyclist. The first time you cycle a route might be awkward as you don't know when you're turning is, which lanes to be in when etc. but the second time will generally be much better. Use an strong or old set of wheels as pot-holes are abundant. Re: Hyde Park: Don't use any cycle lanes that double up as pavement - pedestrians just get in your way. Richmond Park is exceedingly nice.
I thoroughly enjoyed cycling around London as you get to see so many sights, buildings, monuments and views that you miss using the tube. You also get to know where everything is and have a much better sense of the geography of the place. It can also be quite exhilirating despite the smog - but perhaps that's because of the (slight) danger element.