The problem with pinning down the drift of piping traditions in pre-modern Britain and Ireland is that little was systematized until recently - and even now, there are "folk" holdouts that resist being systematized. In addition, piping was not as well documented as we would like in the pre-modern era.

Scotland's Great Highland Bagpipe is the most uniform of bagpipes in its expanse from Ireland to Iran (and beyond). It consists of a conical chanter (playing the melody) and three "humming" drones including a base drone and two tenor drones (which play the same "A" at the bottom of the chanter; the base drone plays the "A" an octave lower). See this image. The uniformity of the Great Highland Bagpipe became essential in the late eighteenth century as they were increasingly played in unison in a marching group, particularly as associated with British regiments. The instruments needed to be manufactured in a way that allowed differed pipes to be played with one another without it producing a cacophony (all jokes about bagpipes aside).

In the nineteenth century, some Irish pipers began to embrace what became known as the Irish war pipes. These tended to have one tenor drone and one base drone, each arranged in the same way as its Scottish counterpart (except there was only one tenor drone). One of the problems with the Irish war pipes was that they were not produced on an industrial level as occurred in Scotland. It was consequently difficult to tune multiple sets so they could be played in unison. Many Irish pipers have consequently defaulted to the Great Highland Bagpipe because it is more easily obtained and more easily tuned together.

There is no question that the Irish had a tradition of mouth-blown bagpipes similar to those played in Scotland. It was and is part of Irish heritage. The question Irish pipers faced was whether to attempt to celebrate that tradition in a modern world with the two-drone Irish variety or to surrender to expediency and adopt to Scottish version that was being mass produced. I have seen one Irish pipe band with the two-drone Irish war pipes (in San Francisco in the 1980s), but it was an oddity that failed to establish a trend.

You are correct to point out that the Irish have what is perhaps a stronger piping tradition with the uilleann (Irish Gaelic for "elbow") pipes. This is clearly distinct from the famous Great Highland Pipes of Scotland, but that does not mean that the "species" of pipe represented by the uilleanns is unique to Ireland.

The piping world is segregated in two ways, the first is in the source of the air and the second is by the nature of the chanter. Air delivery for the Highland Pipes and the Irish War Pipes is accomplished by a mouth "blow stick" - this is internationally the most common approach; the air feeds a bag, usually placed under the left arm. Other pipes are bellows driven with a bellows attached to one arm (usually the right) to drive air into a bag (usually under the left arm). This gives the uilleann pipers a distinct look of having wings flapping like a duck's! This is an image of a uilleann half set of pipes. Here there are three drones playing three different octaves; a full set adds "regulators," which are additional drones that can play chords.

Another distinction is with the chanters. This image features a Highland chanter at top and a uilleann chanter on the bottom; notice the metal "keys" on the uillean pipes allowing for additional notes in a chromatic scale. These represent two importantly distinct species of bagpipe: the Highland chanter is conical. This causes the piper to be restricted to nine notes ("A to A with a G on the bottom"), with the upper notes being quieter than the lower notes. The high "A" - the final note on the scale tends to disappear with the volume of the drones playing the same note two and three octaves below.

The uilleann chanter is cylindrical. The allows all the notes to be played at the same volume, and it also means that a piper can "squeeze" out a higher octave by applying more pressure. Where Highland pipes should be played with steady pressure for consistent intonation, the uilleann pipes thrive on the variation of pressure - making it difficult if not impossible for two uilleann pipers to play together (unlike the Highland pipes).

Conical chanters are most common internationally, but cylindrical chanters do exist outside of Ireland. Because the uilleann pipes combine that feature with the equally rare bellows, it is a rare manifestation of piping. the Highland Pipes represent the most common arrangement when it comes to chanter shape and air source; because of the industry of Highland pipe manufacturing, it has tended to outcompete some local species of pipes. It has consequently, become the standard, not only in Ireland and among Irish Americans, but also in Pakistan and some Arabic nations.

I hope this approaches answering your question. Much of how these traditions evolved - particularly with the nationalist revival movement in Ireland - is disputed and tied up with national pride more than well-grounded history.

edit: Bonus - here is a picture of me playing the Great Highland Pipes back in 1977 in a previous century when I still looked like myself. With this you can see the usual arrangement of playing the instrument. I am playing a low "A" - the right little finger is elevated; if it were depressed, I would be playing the lower "G".

2nd edit: I posted this separately so it would come to OP's attention; I also place it here so it is part of this discussion: I just found this fantastic but unattributed image that is labeled as a Cork IRA Pipe Band in 1918. Most are playin the two-drone Irish War Pipes. Two pipers (front row left - and so probably the pipe major - and center, second from the right) are playing the Great Highland Bagpipe. This is a good example of a pipe band where some pipers had the Scottish-style pipes, while others were able to obtain (or preferred) the Irish pipes. The pipe major is looking to his left, likely as a signal to the bass drummer to signal the end of the set.

I just found this fantastic but unattributed image that is labeled as a Cork IRA Pipe Band in 1918. Most are playin the two-drone Irish War Pipes. Two pipers (front row left - and so probably the pipe major - and center, second from the right) are playing the Great Highland Bagpipe. This is a good example of a pipe band where some pipers had the Scottish-style pipes, while others were able to obtain (or preferred) the Irish pipes. The pipe major is looking to his left, likely as a signal to the bass drummer to signal the end of the set.