This can be a highly technical area. Rather than use a forum, I would say your best bet would be to seek the opinion of a firm of Accountants which would guide through the Tax rules.
If you are resident in Ireland, you will be taxable in ROI on your UK income. However, depending on your circumstances you may be entitled to Trans-Border Relief.
You should take professional advice on this situation.
See borderpeople.info website for useful info on this topic. I'm a cross border worker, living in Ireland but working in N. Irl. Pay taxes in N.Irl but complete an income tax return in Ireland each year. I qualify for trans-border relief.
As each individual's circumstances are different - and some are entitled to Trans-Border relief and some not, there is no straightforward answer. It will all depend on your individual circumstances.