It is now understood you do not fly the original GA (at LHR at least).
Understood by whom, exactly?
PANS-OPS says:
7.4 MISSED APPROACH PROCEDURE WHILE CIRCLING
7.4.1 If visual reference is lost while circling to land from an instrument approach, the missed approach specified for that particular procedure must be followed. The pilot will make an initial climbing turn toward the landing runway and overhead the aerodrome. At this point, the pilot will establish the aircraft climbing on the missed approach track.
This might not be "appropriate" for LHR ATC, but it's absolutely by the book to fly the GA for the instrument approach for which you have been cleared, which is presumably why LHR MATS Part 2 requires an explicit instruction to the contrary.