The cultural and historical reasons for both examples aside,
Well that's a pretty big factor in discussions on subjects like this. Massive, actually.
The correct word probably isn't country, so much as nation. An important distinction that explains quite a bit on its own, no? I dislike generalising about this subject, but its unavoidable if giving a summary. Scotland, Wales and Ireland were all outside the English state at the time in which it was formed and so when they were all (in their own different ways) incorporated into the British state, they were not incorporated as part of England because (of course) there were not part of England; in all cases there was a lengthy period (centuries in the case of Wales and Ireland) between their conquest by the English state and their formal incorporation. It's also worth noting that the two parts of England that were essentially only semi-incorporated until comparatively recently (Durham and Cornwall) have unusually strong regional identities.
That's the first part dealt with. Then there's the issue of identity; how is it possible to have what you call a 'union of nations under another national umbrella'. The key point here is that all four nations were unified (albeit in a rather ramshackle way) as early as 1603; a long time before the development of modern varieties of nationalism.
Where things get a little more complicated is that when Scotland was formally incorporated in 1707 things were different, something even more true when the colony of Ireland was incorporated in 1801. So in both of these nations there were attempts (locally led, it must be remembered) to impose a new British identity after unification (spectacularly successful in the case of the Ulster Protestants and fairly successful - though not lasting - in Scotland). There was never any attempt to do that in Wales (a remote backwater until the Industrial Revolution, without a large city until the middle of the nineteenth century, and with its own language and distinct religious traditions) or in the case of the Catholic majority in Ireland (who weren't even eighth class citizens until Catholic Emancipation), while in England 'British' identity was effectively just English identity given a new word (and even that didn't really catch on until the twentieth century).