Whoopee! At last, a chance to use some of the data I gathered for my degree! 
I can't comment for Wales and Northern Ireland, but I did my dissertation on Scottish names so I can give you a precise answer for Scotland. Basically, the use of Scottish names in Scotland has been visibly in decline for as long as accurate birth records are available (c.1850), and now account for a very tiny amount of names used.
In 1850, 28.2% of boy names and 15.7% of girl names used were Scottish, but by 2013 (when I did the disseration), these figures had reduced to 11.3%for boys and only 6.7% for girls.
This is mostly attirbutable to the size of the names pools, as the actual number of Scottish names in use has increased over the same period, however other names increased even more, reducing the Scottish percentage.
E.g. In 1850 there were a total of 53 Scottish boy names used and 310 'other' names. In 2013, the Scottish names in use had increased to 386 but the 'other' names had increased to 3,024.
So basically, as more and more names become available to choose from, fewer people are likely to choose names from their own culture. Without interviewing parents directly it's impossible to say whether Scots who do choose Scottish names do so for patriotic reasons, or just because they like them, and there is reason to believe that a lot of people don't even know which names are Scottish and which aren't. There are certainly hundreds of lovely Scottish names that became "anglicised" and so fell completely out of use, but I'm not sure how well used they would be even if people knew of them. I suspect the exotic attraction of something different will always trump them. 
quietly tiptoes back to nerds' corner