I lived in central London for 6 years and biked the entire time. I loved it. It's no more dangerous than NYC in fact maybe less so as the cab drivers are much better drivers. You just got to look them in the eye and claim your space on the road.
Go for it!

Richmond Park, from a cycling perspective, is the closest thing you will find to Central Park in London. It is not in Central London, but in southwest London and is easily accessible if you take the District Line to Richmond. Buy an A-Z London map book and you will find it easily enough.

The circuit is 7 miles long and is best ridden anti-clockwise, which gives you left turns only, which are better when riding on that side of the road. Remember that Richmond Park is open to traffic during the day and it is two-way traffic and the roads are narrower. The traffic will be a lot closer to you than you would generally find in Central Park, but it will be much slower and the drivers will be more respectful and curteous.

The park closes to traffic at night time (sundown) and you can still get in and out of the park through the pedestrian entrances. If you have some good lights, you can do the circuit then, but it would be advisable to go with a riding partner. At night, watch out for deer and joggers who will be unlit.

The circuit has hills and flats. The hills are no worse than anything you will find in Central Park. Riding counter clockwise, the northern part of the loop will put you head on into the prevailing southwesterly wind, and this section is exposed and can be quite a challenge. You will have a fine view of the aircraft lining up to go into Heathrow, unless of course the wind happens to be blowing the other way.

There is one steep, short downhill going counter-clockwise that is twisty. It is easy to get up to 35mph and more withough trying, but you have a sharp right-hand turn to negotiate when you are at your fastest on this descent and it maybe adviseable to feather the brakes just before you go into the corner, especially if there is oncoming traffic, and at least until you get used to it. Once you get to the bottom of the descent, deer can be hiding in the bracken to the side and can run out into the road and you can still be going in excess of 30 mph.

If you ride clockwise, take care at the roundabouts (traffic circles). On the northern part of the loop, you will have a fine view towards Central London. Going in this direction, you have a short, sharp climb, which is a bit steeper than the hill at the top end of Central park, but also a bit shorter. The rest of the hills are more gradual.

Beacause it is more exposed than Central Park, especially if the wind is in the southwest, I think the Richmond Park loop is more of a workout, and it is a slightly longer loop.

I agree with other posters that cycling in Central London is a lot easier than in NY and the drivers are generally more courteous. Moreover, the road surfaces are much better maintained than in New York, so you are less likely to encounter potholes. You can find a lot of broken glass on London's streets, though, so don't be surprised if you get the odd flat, or use appropriate tyres. Avoid riding on Oxford Street, it is clogged by busses and taxis and tourists who don't look before crossing the road. The fumes from the buses are nauseating. Use Wigmore Street instead, which runs parallel one block to the north.

If you want to try your luck on Hyde Park Corner (not for the faint-hearted), the safest way to get round is to know which exit you want to get off in relation to where you got on and to move at the same speed as the rest of the traffic. It is a good test of your sprinting skills. But if you have any doubts, don't attempt it, use the pedestrian subways. It's better to get around it in one piece slowly than end up coming home feet first.

If you've not ridden in a country where they drive on the left, take the time to get used to it and to learn which way to look at junctions. Not all junctions in London are controlled by traffic lights.