My experience with onward domestic UK connections via London is limited to Glasgow and Edinburgh. With that disclaimer-

As number_6 says, you will clear Immigration at LGW but won't need to claim and recheck your baggage there.

In Edinburgh, bags that had originated overseas were separated from domestic baggage and diverted to a dedicated "international" carousel. In several journeys, however, I never saw an actual HM Customs officer on duty. I think there may have been a telephone with which passengers were meant to declare dutiable goods on the hono[u]r system. I never saw anyone doing so.