Some of this you may already know
1911 census James Holroyd is shown as born Marsden Nr Huddersfield , Yorkshire. James married Charlotte Elizabeth Hall, Dec Qtr 1873 in Salford, son James William born Mar Qtr 1874 Salford and George Wilfred Jun Qtr 1884 Huddersfield.
James remarried Annie Shaw in 1911.
Frances will confirm this, but I believe formal adoption was only starting on the IOM around 1900, prior to that children were just placed with other families, often after the death of their mother or even father.
I looked for deaths of Wood/Woods around Theodore's birth and found two:
Clara Jane Woods buried Braddan 15/12/1900 aged 32, she was unmarried and was with parents on IOM 1891 census, James aged 61 and Mary 64 who died 1902 (Mary) and 1903 (James) , which may have been a reason to give the child away - long bow.
Thomas Joseph Woods buried 25/3/1900, Malew , aged 31 , no apparent marriage and just possible father.
On " Find a Grave" you can see the Headstone of James, Charlotte and Annie on the IOM.
Do you have Theodore's DC June Qtr 1933, Fylde , aged 32 ? Could be a clue ?
Another long bow , there is a will for Richard Ellis Holroyd from the Isle of Man, since they were the only Holroyd's on the IOM, I am sure he must be related to James, you can view at your local LDS church , reference 63 0106540.