If they were to go airside to T2, they still cross the UK Border at T2 since it's the CTA entry point if going on to the Republic. The desks are on the far right side and it is clearly signed. Effectively the same as going to the Republic via T5, where the UK Border would be in the UK and Ireland Connections hall. Either way they will get the the C stamp, and I doubt the Gardaí are too bothered at their end anyway.

For the baggage reclaim, if the bags don't appear within an hour or two, then yes you can leave an address in Ireland and they will end up being delivered there. You will need to fill in the PIR for this, but BA may text you in advance of arriving into LHR with a hyperlink to do it online.