Technically, nobody needs a passport to travel between the UK and Ireland - doesn't matter where you are from - Mongolia, Mars ... But if you're skin isn't white, you are much safer carrying one. They've got very sticky in Ireland about who they are letting into the country. Also, as I'm sure you know, Ryanair requires photographic identificaiton, a driving licence or passport. Don't know about BA though. If it was me, I'd bring one, or at the very least, a driving licence with photo.
I wasn't asked for my passport when I went to Dublin a few years ago, but you need to carry some form of photographic ID with you when you check in and guess what sort of ID the airlines require....a passport.