I agree, just give it a shot. When you transfer through LHR you will have to go through another security screening and afterwards you have ticketing desks for all major airlines.

But keep in mind that transferring through LHR is a nightmare - last time I did JFK-LHR-FRA I was arriving three hours before LHR-FRA departed and hoped for the earlier flight - turned out I barely made the flight I was booked on.