Milliepede - thanks for the death entry; I might have to bite the bullet and order that!
Mair - well that's put a spanner in the works, but good to know! In answer to your question, there's an Elizabeth Hitchins Hawking, but the rest, bar the two Williams, bear no middle name:
http://bit.ly/2iNNkx3. I also kind find a burial in St Dominick for the first William, so perhaps he didn't die in childhood, which was my first thought. Also, the second William seems to have a fairly consistent spelling of the "Hutchings", with the "s" at the end being the only change, except for his baptism in the name of "Hutchins". On his second marriage entry, his father is listed as Samuel Hawking, which fits with the gentleman in St Dominick because Samuel's occupation is given as "Hosier", and he was a "Taylor" in his will, dying not long after William's birth (although William's marriage entry does not say that Samuel is deceased).
To corroborate things, as you can see in the table, I contacted the Charterhouse School, as he died at the Charterhouse and was a schoolmaster, so I thought he might have taught there. He didn't, but he was a "poor brother", so I have two more consistent entries of his name and something to corroborate the 1868 death. I have written off to the actual Charterhouse almshouse, still in London, to see if they have more detailed records than the school kept, as the school moved just after William's death.
I suppose I won't get any further without something from the Charterhouse or the death certificate now.