Hello all, I may be able to shed some light on this question. I am the aforementioned Harriet's great-great-great grandson. I was told as a child that the Goodman family was Jewish in origin, and that someone in the family converted out of the faith to marry a Gentile; in response, the family hung a black wreath on the door and refused to speak to him/her. I have found them in the 1851 and 1861 census reports from England and Wales, and all the information appears consistent to me. Robert was a coal miner in England in 1851, a collier in Wales in 1861, and a coal miner here in Pennsylvania. However, I did note that their (Robert's and Anna's) birth years are consistent in the census, but do not match their grave marker in western PA. I have also found that there is a Victoria Goodman who resided with Robert and his family in 1851, aged 4, listed as his niece, who then resided with James Goodman in Cromhall, Gloucestershire in 1861, aged 14 and was listed as a granddaughter. I might add that I did come across a possible baptismal record for James that Ancestry had mislabeled as 1786 instead of 1787; to the best of my knowledge, Romani people do not have their offspring baptized in the C of E.
I will add that I have taken several DNA tests. 23andme produced an absolute puzzle for me- it listed Iberian and ~.1% Middle Eastern/North African DNA, neither of which I can explain. From what I've read, British Travellers have some genetic contribution from each of these populations. However, I also (unsurprisingly) had .1% Ashkenazi among my results. These results may be just 'noise', or they might show that the Goodmans were, in fact, Romani.