I know original poster says he is not checking bags, but to others who might read this thread: in my experience it is NOT possible to check bags all the way through a LHR-LGW transfer. I was told the opposite by an AA baggage attendant at LHR, so I left the airport, went to LGW, and was then told that my bags would not be transferred. Bags were lost and did not show up at my destination for three more days.

BTW, taking a taxi for this transfer is RIDICULOUS. Take the coach -- it's very easy. But do allow ample time -- 90 minutes MINIMUM, perhaps more if during rush hour.