As best I recall, the UK position is not governed by a specific law. Rather, there is case law to the effect that fast growing dense Leylandii are capable of constituting a nuisance, and that you can therefore require that the nuisance be abated. ( I suppose you could say there is a specific piece of case law). I am unaware of any Irish case law, but UK court decisions would be regarded as having some persuasive power before an Irish court.
You need to talk to a solicitor, and the chances are that the solicitor will need to get a written opinion from a barrister. This does not sound like the sort of case where you will be told that you have a highly winnable case. At best, they will probably tell you that there is a stateable case. Certainly, any good solicitor will urge you to try to resolve this without litigation.
If, following detailed advice, you decide to bring a case, the chances are that it will cost at least €5k. Even if you win, the costs awarded may not cover your full legal bill. If you lose, it could end up costing you in excess of €10k.
Please get specific written advice before proceeding (and under no circumstance should anybody rely on my observations here - my observations here are intended only to prompt people to get written legal advice, and in no degree as a substitute for it). However, with that caveat, you are entitled to cut away both branches and roots to the extent that they protrude into your garden. If a tree relies on roots in your ground, if removing these roots would destabilise a tree AND if it it would fall safely (i.e. not onto a house or person) then you might consider taking appropriate action (under the supervision of a tree surgeon).