From BA:If you are a national of the UK, including the Isle of Man, Guernsey or Jersey, or a national of the Republic of Ireland, you do not need a passport to travel between these destinations, but you do require some form of photographic identification that proves your identity and nationality.

All other nationals require a valid passport to travel between these destinations to confirm your identity and nationality, with evidence of your immigration permission, if applicable.

https://www.britishairways.com/conte...-visas-and-api

You shouldn’t have any issue with Irish immigration. Additional info here: https://www.citizensinformation.ie/en/government-in-ireland/ireland-and-the-uk/common-travel-area-between-ireland-and-the-uk/