As both the UK and the Irish Republic are within the EU, the police (or, more accurately, the Crown Prosecution Service) can apply for a European Arrest Warrant to have the offender arrested and returned to the UK.
Unlike the long-winded extradition processes which apply where non-EU countries are concerned, the EAW process can be remarkably swift. Once a British judge has been presented with sufficient evidence to show that there's a prima facie case against an alleged offender, he can issue an EAW which is then forwarded to the International Crime Bureau (which is part of the National Crime Agency). They, in turn, send the warrant to the relevant country/ies, who are obliged to arrest the alleged offender and return him to the UK. (Unlike extradition from a non-EU country, there's no long-winded appeals process in the country where the alleged offender is living. He must automatically be returned to the UK if an EAW has been issued).
Further, it's possible that the Republic of Ireland's own legislation might grant its own authorities extra-territorial jurisdiction, meaning that the offender could be tried and convicted in Ireland. (The UK's own Fraud Act 2006 permits that people who've committed fraud abroad can be tried, convicted and imprisoned here in the UK).