It’s a bit complex!
The Church of Ireland is not Presbyterian, it’s Anglican. It’s by far the biggest Protestant Church in the south, but in the north the Presbyterian Church is significantly bigger.
The Church of Scotland is Presbyterian. But there is a small Anglican church in Scotland, the Scottish Episcopal Church (whose adherents are sometimes called “Piscies”). The Church of Scotland is the National, but not the State Church (as the Church of England is, in England - but certainly not in Scotland). When the Queen is in Scotland she attends Church of Scotland services.
The Church in Wales (“in” but not “of”) is the Anglican church in Wales. It is neither the national, nor the state, nor the established church. Like the Church of Ireland and the Scotish Episcopal Church, it is an independent church in the Anglican Communion. Rowan Williams was previously Archbishop of Wales. He is, incidentally, not English, but Welsh born and bred and a Welsh speaker.
Whilst writing, you might know that the RC Church has separate hierarchies for Scotland and for England & Wales (the RC Church in Wales comes under England).
On Ireland neither the RC Church nor the Church of Ireland recognise the border between the north and south in their (almost identical) structures. Some dioceses and even parishes cross the border. There is just a little tension on occasion between north and south in the Church of Ireland.