I've been to London several times and I honestly never had the desire to cycle there. It's congested and unlike newer cities that may be laid out in a grid, the streets are twisty and haphazard. Maybe the parks?

I have ridden in the English countryside. There are organizations that may help you plan a day out. Take the train somewhere and set you up with bikes and maps.

How long will you be staying? And is it just a vacation, or is business involved? If it's going to be for a while, I would try to find a more residential neighborhood (probably better biking) and rely on the "Tube" to get to the sites.