GC lounge at LHR has moved to gate 88 recently; not sure why. I don't know if a BA card will get you in there (since EI left OW some time ago), but doesn't hurt to ask.
BA codeshares with EI on DUB/LHR but the BA flight number is sold only when part of a multi-sector journey. Ditto ORK, BFS, SNN to LHR.
H class on EI doesn't earn BA points (despite that it's EI's second-highest restricted economy fare bucket); see
here for more details.