This is a good song and is adaptable by region ...
Edinburgh : She is the girl from the golden city Belfast : she is the belle of (the) Belfast City
I always sing Jenny Murray.
Jenny Murray and Albert Mooney sound like Belfast names but I suppose they could be Scottish too.
I am orig. from Belfast and there were Mooney's and Murray's in my school.
I think the following adaption is peculiar to N.Ireland.
My Aunt Jane, she called me in, She gave me tea out her wee tin, Half a bap with sugar on the top And three wee mice from her wee shop.
(A bap is a large round bread roll, the mice are sugar mice )