Yes and no. (According to the way I’ve been taught, at least)
In everyday speak, they are seperate countries bonded by shared legislation. It’s proper to refer to a scotsman as being from Scotland, the country, in a way it’s not proper to refer to an American abroad as, for example, a Texan, Iowan or New Yorker. He’s Scottish and she’s American, living in New York, if you will.
However, in formal and political context, all four countries are considered nations that, together in the alliance, constitute a country.
To the extent necessary for this particular discussion, sure. They may not be nations in the same way as, for example, the United States, but there are plenty of people in Great Britain who consider themselves English or Scottish or Welsh even more strongly than they consider themselves British.
And if you accept that a nation is, as Benedict Anderson famously wrote, an “imagined community,” then i think that having a national anthem and a sense of national community is probably sufficient, at least for our purposes.
I should point out a pet opinion of mine regarding national anthems. In both Australia and America (and probably elsewhere too), there is a lot of feeling that “this here other song” is better, more well-loved, and should be the anthem instead.
I say that’s codswallop. When I am overseas, it is “Waltzing Matilda” that will bring a tear long before “Advance Australia Fair” ever will, and I also think “America the Beautiful” is leagues ahead of the SSB…
…but…
…if we made those songs our new anthems, they would lose something special- instantly. National anthems are for dull events of state, politicians, and endless, boring schooldays. Let the ownership of the “real” song stay with the people. “Waltzing Matilda” is a hundred times better than “Advance Australia Fair”, but if it ever goes to another vote, I’m voting it down. That song doesn’t belong to the government, it belongs to me.