If separated people have new partners, is it "adultery"?

Adultery is not a crime.

Have you an example if those state firms?

Can I suggest we stop using terms like 'crime' (OP) and 'offence' (above) to describe sex between consenting adults who were not married to each other?

Criminal conversation was a tort, it was never an offence. The remedy was, as you pointed out, in a civil case.

There is no crime of adultery in Irish law. And it is extremely unlikely to have any effect on divorce proceedings - Ireland operates a no-fault divorce system: https://www.listonfamilylaw.ie/affairs-in-divorce-proceedings-in-ireland/

As others have pointed out, adultery was never a crime in Ireland. Back in the day, though, it was a "marital offence", meaning it was one of the grounds on which you could seek a judicial separation or, in England, a divorce. (The other marital offences were cruelty, desertion, wilful neglect.) The marital offences are no longer relevant, since we have a no-fault basis both for judicial separation and for divorce.

But, to answer the OP's question, back in the day when it had some legal relevance, if a married couple were living apart and one of them had a sexual relationship with a third person — yes, that was the marital offence of adultery and was grounds for seeking a judicial separation (or, in England, a divorce). And in those times it would have had an influence on court decisions about maintenance, alimony, etc.

In fact, under the English divorce regime, one spouse could seek a divorce from the other on the grounds of the other spouse's adultery. But, bizarrely, if the spouse seeking the divorce had also committed adultery, the divorce could be denied on the basis that that spouse's conduct justified or excused or at least mitigated the adultery of the other spouse.

But all this is in the past. Nowadays if separated people embark on new relationships it's still adultery within the dictionary definition, but it has no legal consequences.

Separated people embarking on new relationships isn't really an instance of polyamory, thought.

Probably wise, then, not to try and bracket it with adultery, which is socially deprecated because of the deceit, dishonesty and betrayal commonly associated with it. Advocates of polyamory would presumably say that it is the opposite of adultery, rather than something to be linked with adultery.