How is religion tied to ethnic identity in the UK and Ireland?

I can speak from experience that in Eastern Europe, religion is tied to your ethnicity.  For example, in Eastern Poland if you are Eastern Orthodox, Jewish or Muslim you would not be considered to be ethnically Polish, even though you might actually be or claim to be.  Polish people would not accept you as one of "them".

So in the NE areas of Poland, you can speak Polish, claim to be Polish, your ancestors are from that area but if you are Eastern Orthodox then people would consider you to be Belarusian or Russian.  If you are Muslim, you would be considered Lipka Tatar.  If you convert to Catholicism, then you also "become" Polish in people's eyes.  

In the former Yugoslavia you cannot be Serbian and Catholic, you would then not be considered to be really Serbian.  Only Croatians are Catholic.  If you are Muslim then you are Bosnian and of course if you are Serbian you have to be from an Eastern Orthodox family.  If you are non religious or atheist people consider your previous religious affiliation or that of your family to assign you to an ethnicity.

So in the UK, if someone in Belfast identifies as Ulster-Scots but they are Catholic, would they be seen as Ulster-Scots or as Irish?  Conversely, if you live in the Republic of Ireland and are from a Protestant family, would you truly be recognized as being Irish?  If you change religion does your ethnic affiliation change?  How about the rest of the UK?  A Catholic Englishman or Welsh person?