Hi,
You would need to register for self assessment, as you will be in receipt of foreign employment income, which you would declare in the tax return.
You should also request a certificate of residence (
https://www.gov.uk/guidance/get-a-certificate-of-residence) to send to the tax authorities in Ireland. This will allow them to tell your employer not to deduct income tax in Ireland.
You should also contact the national insurance enquiry helpline for advice on national insurance.
Their number is 0300 200 3500.
Thank you.