With six hours for the connection, I would not only go through U.K. Immigration, but I would only check my bags as far as LHR, and claim & re-check them myself. This removes an entire layer of uncertainty.

Your connection is probably from T-3 to T-4; after you get your bags you can take the Heathrow Express downstairs for free to Terminal 4, check in at BA, and then relax. You might even find an Internet access place!