It is possible to serve in the national legislatures of the UK and Ireland at the same time, although only one person has actually done it. Several other people have been in both, but not simultaneously. The overlapper is Benjamin Guinness, the third Earl of Iveagh, who was chairman of the well-known beverage company. He was a member of the House of Lords from 1967 until his death in 1992, by virtue of his peerage title. He was also a member of the 13th Seanad Éireann (1973-1977), nominated by the then Taoiseach.
There are some differences in the rules relating to the different chambers in both countries, and to subnational legislatures like the Northern Ireland Assembly. As to the UK, for example, someone is disqualified from the House of Commons if they are a member of another country's legislative body - unless that country is Ireland or a Commonwealth country. And Ireland is the most likely other country to be involved since there are many dual citizens and it is physically closest; being a UK and NZ double-MP would be inconvenient.
Being a dual citizen does not disqualify you from being a UK minister. Recently, Boris Johnson was a US citizen (as well as a UK one) while Foreign Secretary. If he were elected to office in the USA then he would have been disqualified from his Commons seat. There is no such legal bar in the UK if he had been appointed to some important American job, such as being Secretary of Commerce - assuming US law allowed that. It just becomes extremely politically unlikely that this sort of thing would happen.
Although not a political office, it is also possible for senior judges to hold office in more than one country. Until recently, Lord Reid and Lord Hodge were on the UK Supreme Court as well as the Court of Final Appeal for Hong Kong. Several other CFA judges are also current members of the UK House of Lords. Previously, Lord Cooke was a New Zealand judge who also sat as a judge in the Lords (prior to the Law Lords being split off into the Supreme Court), as well as on the CFA and other courts of Pacific nations.