The quick answer is that Norsemen, Gaulish Celts, Romans (via Egypt to Spain [where some of those nationalities were collected] to Ireland), and the original indigenous peoples make up the ancestry of today's Irish, Scots, and Welsh.
It's kind of a generic answer without a lot of details. There were several waves of migrations that took place around the B.C./A.D. timeline. There are few, if any, reliable record of the indigenous island peoples living in Ireland and Scotland, especially since they intermarried with the newcomers and basically disappeared. The Scots were supposedly named after Scotia, the name of the Egyptian princess who married an exiled caesar's son. They left Egypt and traveled up to Spain looking for a country to call their own. That didn't work out and they set sail for the area we know as Ireland today. Eventually, they spread to the Scotland area, where it is believed that some Norsemen already lived. The Celts were already living in the area.
The Scotians moved back and forth between Ireland and Scotland until they eventually separated into distinct superclans on both islands. Wales was a part of these migrational moves. Ultimately, however, Wales was conquered by the English.
This is a really broad brush approach to the heritage of that area without a lot of details. The history is rich and fascinating and is very well recorded by monks once the celts and the Scotians (later shortened to Scots) conquered the land.
The Celts
Welsh
Irish: dóchas Scots Gaelic: dòchas Manx Gaelic: doghys, jerkallys, treisht Welsh: gobaith Breton: spi Cornish: govenek
Irish Gaelic, Scots Gaelic, and Welsh are all Celtic-Gaelic languages that are still widely spoken today. There are also Cornish and Manx which are now mostly dead.
The 'name' Wallace is an old Scots term for Welsh speaking or 'of Welsh stock'. It is thought that his ancestors were certainly Welsh and they ended up living in and owning land in Elderslie near Paisley. The Wallace's left Wales to set up home in Scotland around the year 1170 but as William was born around the year 1270, it is very likely he was born and raised in Scotland and was 100% Scottish.
Irish, Scottish, Welsh
Welsh, Scottish, or Scots-Irish
Yes, Brad Pitt does have Irish ancestors. He was born in Oklahoma, and has ancestors from England, and has smaller amounts of Scottish, Welsh, Irish, and German.
Welsh, Irish, Scots Gaelic, Breton, Cornish, Manx
Irish: aingeal Scots Gaelic: aingeal Welsh: angel
Irish: creid Scots Gaelic: creid Welsh: credu/coelio
In Irish: In Scots Gaelic: In Manx: In Welsh: In Breton: In Cornish:
Irish: Scots Gaelic: Manx: Welsh: teulu Breton: Cornish:
Irish: blas, tástáil Scots Gaelic: blasad Welsh: blas; chwaeth
In Irish it's "gaoth" In Welsh it's "gwynt" In Scots Gaelic it's "gaoth"
The Irish for Frost is Sioc. Scots Gaelic is reòthadh or sioc Welsh is rhew
Irish Gaelic: tarbh Scots Gaelic: tarbh Welsh: tarw
Irish Gaelic: coinín Scots Gaelic: coineanach Welsh: cwningen