He lived with Fannie (Bellingham) Cheeseman and Hester Bellingham in Newark, NJ, (Fanny is definitely child of Jane McCart), so he would appear to be the son of Jane McCart and Thomas Bellingham.
BUT - he named fifth child Agness (but only his second son, Thomas), so he might be the son of Agnes Dempsey and Thomas Bellingham. He could be a half brother.
In March 2008, Al Luce writes, "I have just returned from a two day trip to Trenton, New Jersey with some news about the Bellingham's of Newark, NJ. It is my firm opinion that James Bellingham was the son of Thomas and Jane McCart Bellingham. As evidence I present:
1. The name of the second female child was Mary Jane Bellingham (the Mary may have been an effect of his marriage to a catholic widow with a son and a daughter).
2. The Agnes that we found in the 1870 and 1880 census was an adopted daughter (her 1881 baptism identifies her parents as David Jameson & Martha Smith) and the original marriage record for them adds the fact that Martha Smith's mother was Agnes Smyth."