Thanks for all your helpful suggestions, James, particularly regarding the technology side of things which I'm completely useless at.
Here is a link to the poster for anyone who would like to print it off or share it:
http://i.imgur.com/ZEhzHsm.jpgApologies for not providing full information in my original post - there's just so much to report that I didn't want to overwhelm readers, particularly when many of the leads we've explored have turned out to be dead ends.
Yes, we got Patrick's birth certificate. Angela gave a false name on it - Mary McGowan - which combined her first Christian name (she was Mary Angela but went by Angela) with her mother's maiden name. This delayed the search for a long time. She gave her address simply as 'Crumlin Road, Belfast'.
Angela travelled from Belleek to Dublin in December 1944, when she was 6 months pregnant. She stayed at the Legion of Mary Regina Coeli Hostel and gave birth to Patrick at the Rotunda on 16 March 1945, before leaving for Belfast with him on 24 April 1945.
We also have Angela's birth certificate and baptismal record. Her full name was Mary Angela Dolan and she was born on 5 June 1918. Her family came from the town land of Killybig near Belleek but her birth was registered in Ballyshannon, just across the border.
Angela used the name Mary McGowan again on the papers for Nazareth House when Patrick was placed there in 1948, but this time she gave her address as 14 Eglinton Street. We believe this to be a false address because Eglinton Street was more or less destroyed in the Blitz and street directories show that numbers 2-46 were vacant during the years between 1945 and 1948, when Angela claimed to have been living there.
I am now investigating the possibility that Angela stayed with a relative or a family friend somewhere around the Crumlin Road, since her use of the Eglinton Street address implies some level of familiarity with the area. Between the years of 1945 and 1948 Patrick was fostered by a woman called Mrs Larkin who lived in Whitehouse, Greencastle, and in my opinion this adds to the theory that Angela lived in the North of the city during her time here. My next step will be to consult the street directories for the Crumlin Road and its surrounding streets to see if I can find any surnames that can be connected to the extended Dolan family, since I imagine that if she did go to stay with someone there, they were probably a relative or connected to one of the local families in Killybig. Relevant surnames are Dolan, McGowan, McCann, O'Dare, McGee, Branley, Rooney and Monaghan.
The local rumour in Belleek is that Angela emigrated to New York to work at her aunt and uncle's restaurant in Yonkers. There is some evidence to support this: the name Mary A. Dolan does appear on the passenger list of a ship which sailed from Liverpool to New York on 6 May 1949, one year after Angela placed Patrick in Nazareth House. This Mary A. Dolan would have been the right age - 30 - and gives her address as 5 Scotland Street, Belfast. Unfortunately, it appears that the Immigration and Naturalization Service accidentally disposed of the manifests for ships that arrived at New York between 1 January and 17 September 1949, so there is no record of her arriving in New York, and the passenger list for the Britannic does not show the address she was travelling to. The CMT also contacted descendants of the aunt and uncle in Yonkers and they had never heard of Angela.
I and a few others have trawled through the records on Ancestry.com and ruled out all of the Angela Dolans, Mary Angela Dolans and Mary A. Dolans there (as well as those Mary Dolans whose birth year roughly matched Angela's - there are so many thousands of Mary Dolans that investigating them all individually is a pretty impossible task). We have also found nothing in PRONI's records. Over the past two and a half years, the CMT have made their way through US registers of marriages and deaths on a state by state basis. I don't know exactly what databases they used, but I know they are fairly certain that they have exhausted the American leg of the search. Other websites I have checked are the Liberty Ellis Foundation, Find My Past, FamilySearch and RootsUK.
Hopefully that helps to give people a fuller picture of what has been done already. I should add that I may know who Patrick's father is but I don't feel at liberty to post any information about him and, at any rate, none of his living relatives have ever heard of Angela. Please feel free to ask if I've left anything out of this post.