Did Annie Louisa marry George Ellis Metcalf(e)?
If so then Harry/John Thomas signed up as Harry Harold Baker for WWI - and the file is marked to say his sister is his only living relative, so likely he didn't know anything about William either.
http://trove.nla.gov.au/ndp/del/article/52329398/5063642?searchTerm= - from here, they were in Emerald.
http://trove.nla.gov.au/ndp/del/article/52330378/5063651?searchTerm= here in 1890 Alfred Baker was up before court, charged with child desertion. He was discharged (supposed to pay 12s a week for their support), and the police magistrate was going to try and see if his wife couldn't be discharged from the "reception house", in which case if he took her away and provided a home for her, he could have his children back.
From the records from the Rockhampton orphanage, it doesn't appear that this happened - Harry was discharged in 1894 "to mother", not "to parents", so I think it's more likely that Emily came out of the asylum by herself and took him back, and Alfred may never had been back in the picture.
I went looking for the English records to see if that gave any hints - the marriage is, I think, at freebmd.rootsweb.com, in 1883 in Woburn district which is indeed Bedfordshire, but Emily's maiden name is indexed as "London" rather than "Lennon".
https://familysearch.org/pal:/MM9.1.1/N6JG-GJ6 - also here, giving Alfred's father's name as Alfred and Emily's father as Thomas.
The nearest set on censuses are Alfred son of Alfred and Charlotte Baker nee Bradford (aged 10 in 1871, 20 in 1881 indexed as Barret living in Battlesden where he married).
