Er, no?
Your first sentence is heading in the right direction: if you'd been out working full time for a complete tax year, then your earnings would normally escape UK tax. However, you haven't satisfied this test, and you're taxable in the UK on your worldwide earnings, as and when you earn them.
I suggest you d/l
http://www.inlandrevenue.gov.uk/pdfs/ir20.pdf
and have a look at the parts 2 and 5, and 6 if you're considering bank interest.
I am struggling to see how the remittance basis applies. Note the issue of domicile is not relevant where the employer is resident in Eire.
I should think that you apply normal UK tax law to the overseas earnings: you are deemed to earn your employed income on the earlier of when you are paid or it is contractually due to you, and you should apply the exchange rate that was in force, presumably on your normal pay day for each salary payment made.
Hope this helps, but I am not sure it is what you wanted to hear.