This is a good question. Let me give you an exemple:
If I fly on a ZED ticket HKG to NRT on CX then NRT to LAX the same day with my airline I show them the CX boarding pass and the agent override the depature taxes. I do not know how it works the other way around.
If I was you I would ask the BA gate agent (and let us know), you have nothing to loose.
The ridicoulous taxes in Great Britain makes me avoid this country plus I hate LHR or LGW (but it is personal)
Have a good trip.