Thanks for the answers!

I know that in the past it was possible to transit through the UK with a Russian passport without a visa for up-to 24h even if it required a ground segment (e.g. the airport change). They changed it a few years ago.
So I believe the safest option related to the luggage would be to check both bags on my name, then I can re-check them in London because I hold a French passport.

One more question related to this: can BA or IB agents merge two separately purchased tickets into one PNR? This will solve all our problems. My govt travel agent can do it at the time of the booking -- I used to have AA + LA or even DL + LA on the same PNR (the latter combination usually yielded weird consequences), but I'm not sure how easy it is to do after two tickets had already been purchased -- I'm pretty sure it is but the question is whether or not they will be willing to do it.