Hi Ewan , looking at the Cox's , appears that John Brotherton ( Also listed as Bretherton and Bulhuton) Cox and William Henry Cox were brothers, from the IGI :

John Brotherton Cox
Born 1/12/1838 , ch 2/10/1840 , St Phillips, Birmingham, Warwick, England

Father John Brotherton Cox
Mother Eliza

William Henry Cox
Born 6/4/1837 , ch 1/5/1837 , St Phillips, Birmingham , Warwick, England

Father John Cox
Mother Elizabeth

John Bulhuton Cox married Mary Jane Allport
St Barnabas , Douglas, IOM 24/4/1877

And from Brian Lawsons births had a least 4 children 1878-1887 .(name shown as Bretherton)

On the free 1881 census he is shown as a Boarding House keeper.

On Brian Lawsons burials William Henry Cox is shown as being buried 4/11/1910 in Marown aged 77 , is this him ? His age seems to be different every time it is recorded. Have you asked Donna Douglas if there is a M I ?
This site also shows William Gordon Cox being buried 22/8/1885 in Braddan, aged only 4 months, cant find this child being born !!

Cheers
David