Hello Ewan,
A couple of thoughts:
Lionel H aged 1 in 1881 was Lionel Harry, birth June quarter 1879 Plymouth (5b 293). As he got older he might have been known as Harry?
By the way, did you see that the National Portrait Gallery has "Sheila (née Field), Lady Tyler, Second wife of Sir Leslie Norman Tyler; Widow of Lionel H. Cox; daughter of Eric Athelstane Field"? Sitter in 2 portraits (1961).
Secondly, there was an Alice Edwards living in the household (daur aged 17) where Gordon was a boarder in 1891. Could this be the Alice who was living as William Henry's wife in 1901? As his wife was still alive that year I don't think they would have married.
Thirdly, when he married in 1877 was William Henry a widower? And what was his father's name and occupation? His age given was a bit inconsistent, and I wondered whether his birth could have been as early as 6 April 1837, and bapt 1 May 1837 at Saint Phillips, Birmingham, Warwick, son of John COX & Elizabeth?
Sue