Monica
I have checked William's death record and he was 78 when he died in 1918, so his birth has to be close to 1840.
On a previous check of the 1851 census I found the Margaret you mention, but I think William is too old, however I also found Jean Bryden (widow) with a son, William age 11, living at Summerhill in Hollywood. What is interesting is that her husband (whoever he was) must have been quite a bit older as there is a stepdaughter (age 28) at the same address.
The problem is there is no way that I can think of confirming these as my relatives.
With regards to William's children, unfortunately that is no help with names like Harry, Walter, William and Louisa, although the eldest son's middle name was John.
I had previously checked the Brydone name in various trade directories for Liverpool and there were several people living in that area with that name, so I think that name's a no go.
Mike