Yes, if you're going to Windsor, I'd also recommend that you find a Windsor-based taxi company to pick you up. I don't have any specific knowledge, but a quick Google search suggests that the fare would be around £18 each way.

If you want to go into London, and particularly to Westminster, the easiest way (even if it can feel quite slow) is on the Tube - change at Barons Court for the District Line to Westminster. There's plenty double decker buses around, but only route 15 (well, part of route 15) is operated by the heritage routemasters (the traditional, open-backed old double deckers). These go from Trafalgar Square (a short walk from the Houses of Parliament) to Tower Hill, taking in the Strand and St Paul's on the way, taking about half an hour to do so.

From Tower Hill, you can reverse the journey - take the District Line (NOT Circle or a Wimbledon train) to Barons Court and change to the Piccadilly Line to Heathrow. Alternatively, you could get off at St Paul's, walk to the Millennium Bridge, and return from Blackfriars.