OK, the position here will be, I think, that you will pay UK tax, national insurance, etc in the usual way — the employer will deduct it. The profit-share bonus from the EOT will be paid without deduction of tax, since it's tax-free in the UK.
You're liable to Irish tax on your income from the employment (including the profit-share bonus) but you get a full credit for the UK tax you have paid. You submit a tax return each year and pay whatever amount is due.