LaVolcan, there aren't really private schools in NI, as far as I know there is only one fee paying school. Although you can attend the 'voluntary' grammars as a fee paying student, but in reality very very few pupils do, the vast majority are not fee paying.
It is also not the case that there are Protestant schools, there are state schools and Catholic grammars, but not Protestant grammars, although it is of course the case that by default the majority of pupils at most state grammars are from a Protestant background. But some grammar schools in an area where there is no Catholic grammar will have a fairly mixed intake. And there are integrated schools which actively seek to have a fairly evenly balanced intake.
There are not very many comprehensive schools, although they are more prevalent in some towns than others. Afaik, there are more Catholic grammars than state grammars, with Catholics making up a smaller proportion of the population. I'm not sure how that corresponds to actual places available at grammar schools, as obviously they will have different sizes of intake, but I'm guessing that it might equate to having a wider choice of schools available, since all the state schools are also available to Catholic pupils too. In the area where I live, the most oversubscribed school is the one comprehensive. It is the only comprehensive in about a thirty mile radius, but its also the only integrated school in the same radius, so its not clear if its popularity is because of its integrated nature, it's comprehensive nature, or a combination of both.
I have no idea how the education system works in the rest of the UK, but what it really great about here is that there is none of this crap about your parents having to be able to afford to live in a particular area in order to get you into a decent school. Selection at 11 isn't perfect I admit, but it's a lot fairer than getting stuck with a lesser school on the grounds of your address. And also, the non grammars have often got very good reputations too. Many non grammar pupils go on to university for example.