Am assuming the EI segment is a codeshare and has a BA flt number? That being the case you'll be checking in with BA (or it's agent/handler, which may well be EI at Cork) for a BA flight ORK-LHR-HKG and should receive both boarding cards and have your bag checked all the way to HKG.

I haven't done that from Cork but have done it from SNN where the check-in was with EI and from DUB where the check-in was handled by Swissport.

The worst case I've experienced is that they tell you (because of some systems issue) to pick up the second boarding card at LHR but even then they have always checked my bag to the final destination.

Hope that helps.