Admittedly it's circumstantial.
All the addresses for the army records are associated with the family of John and George Blackham, so it would be a huge coincidence if it wasn't a very close relative (even a cousin which there isn't anyway) and the final address is where John Blackham is living in 1921.
John's lie was simple - his name - he didn't marry until after he was discharged (in 1919). What checks would be made to make sure someone was telling the truth when they enlisted? Enlisting in a different regiment what chances were there of being found out? The drink problem would be John's.
George doesn't disappear, he emigrates to Canada in 1912 where presumedly he dies. He is followed by sister Ethel and then John. The discrepancies on George's marriage to Laura could be due to his brother using his name (sorry speculation again).
It is a lot to accept so your doubts are understandable, my opinion is ultimately irrelevant.
A couple of things might help to eliminate George as being the person in the Worcs - his death record in Canada which usually lists both parents' names, unfortunately it is after what's available, or his birth certificate which should give his dob to compare with the emigration records.
One other thing is if both enlistment forms are signed you could compare signatures to see if they match, no idea how easy that would be.