I've found out that there is a contradiction in sources and am not sure which is right. I now have two different alleged lines of descent as follows:
11. Thomas Bambrough + Ann Hood b 1669
10. Thomas Bambrough 1703 + Margaret Nixon 1705
9. Thomas Bambrough 1721-1800 + Margaret Jennings 1725-1762
8. John Bambrough b 1754/1755 + Isabel Forster of Fellside 1753 (m 1755/1781?)
7. John Bambrough 1786/1787/1788/1789 + Dorothy Storey (m 1817)
6. William Bambrough 1820 + Elizabeth Beylass
5. William Bambrough 1851 + Ann Swallow (m 1873)
4. James Bambrough 1887-1918 + Margaret Beatrice Newman 1886-1928/1931 (m 1908)
3a. William Ernest (Laurence) Bambrough 1910-1984 + Ruby Constance Wildermoth 1911/1912-1979 (m 1934 NZ)
(3b. William Ernest Bambrough + Anne Patience Mcleman 1931-1984)
(2a. K... G... Bambrough (my father) + J... A... Wright)
2b. K... G... Bambrough (my father) + Jennifer Mary "Edwards" 1946-2020
1. S... W... Bambrough (me).
8. John Bambrough 1753-1833 + Elizabeth 1765
7. William Bambrough 1794-1876 + Margaret Thompson 1801-1863 (m 1819 Durham)
6. James Bambrough 1825-1888 + Dorothy Heslop 1830-1900 (m 1850 Durham)
5. William A Bambrough 1851-1916 + Ann Swallow 1850-1904 (m 1873 Durham)
4. James Bambrough 1888-1918 + Margaret Beatrice Newman 1886-1931 (m 1908)
3a. William Ernest (Laurence) Bambrough 1910-1984 + Ruby Constance Wildermoth 1911/12-1979 (m 1934 NZ)
(3b. William Ernest Bambrough + Anne Patience Mcleman 1931-1984)
(2a. K... G... Bambrough (my father) + J... A... Wright)
2b. K... G... Bambrough (my father) + Jennifer Mary "Edwards" 1946-2020
1. S... W... Bambrough (me).
Which just goes to show I have to be more careful with my sources etc. Unfortunately I am not able to go buying all the birth certificates etc which makes it less easy to have proof of parentage etc. The second one version is from Mormon site.
(In the first version I had difficulty finding out the surname of Isabel of Fellside. There were three records of Isabels of Fellisde marrying a John Bambrough in roughly the same year. Isabel Archer, Isabel Maugham, Isabel Forster.)
I still can't see/find any "Prussian prince" or "German prince" anywhere in my grandfather's ancestry, nor any marine engineer, as supposed in our family tradition. I think it must be wrong/fictitious. Except that Shauny Bambrough from Northumberland said they have the same prince tradition in their family too, which makes me think/feel there must be something to it. My father said his father told him we are descended from a "Prussian prince" or "German prince" via a marine engineer. I have thought maybe "German" was a mistake for Germanic/Teutonic Anglo-Saxon Bernician/Deiran Northumbrian Angle, but it doesn't seem likely. The genealogical sites only seem to go back to 1600s and not any earlier. Recently I wondered if my grandfather's middle name Ernest might point to the Saxon/Thuringian Ernestine Wettins but I can't find any link. About 10 years ago I met at a group meeting a lady here in NZ who once came from the UK, and when I mentioned the Prussian/German prince story she thought it might be the dwarf Polish Count Joseph Boruwlaski who was in Northumberland, but I can't see any link there.
I once found in google search a page in a genealogical site with photos etc with some woman who married a Bambrough which looked like it might be the German/Prussian link but I can't remember her name and can't find it searching again, and since google changed their search to worse than it was before I can't find it in my search history. I think she had some name like Isobel or Lutetia or something.