And Churchill to Picadilly Circus is about 20 minutes walking. I'd expect it to be cheaper than the trip to Paddington except there is often a lot of traffic on the main streets if the taxi chooses to take you that way (and it's difficult to avoid for at least a small part of the journey on Regent even if the cabby wants to).
Churchill to Knightsbridge, probably also 6-7 pounds and traffic shouldn't be too bad. St. Pauls would be more.