You'll need to change terminals, but they are only a few minutes walk apart.

EI checkin closes 45mins before departure, so adding a 5 minute margin means arriving at the EI checkin desks by 9am.

If you are checking in luggage, you will presumably need to wait to collect your luggage from BA on arrival.

If the BA flight is on time you should just have enough time to collect luggage and checkin with EI.
If the BA flight is delayed, or luggage is delayed such that you miss EI checkin then you'll potentially wave goodbye to your ticket.

If you can avoid checking in luggage you'll have a much better chance. If you're connecting at LHR to a BA service same day, you can just check your luggage in at LHR for the longhaul segment rather than LON-DUB, collect, check DUB-LON-XXX.