Okay, so if he was taken back to St John's Chapel, Durham Records Office has the parish Records for St John The Baptist, St John's Chapel
They have burial Registers that span 1788 - 2011
Given you have the date of death, I should think looking through a couple of weeks maximum would confirm if he was buried there. As its a parish register it may not tell you the section and plot number but it will at least confirm if he was buried there and when.
http://www.durhamrecordoffice.org.uk/Pages/AdvancedSearchChurchRegistersDetail.aspx?SearchType=Param&SearchID=36c872d7-e0d2-4de0-a945-f94424e28a9e&ItemID=597875Note the reference numbers, if you can't actually go to look for yourself, give them a ring and (for a fee) they will find and copy the record for you.
I'd also Google the funeral directors, 'if' you can track them down they 'may' have records that can tell you the section and plot and whether or not they supplied a headstoneGoing back to him being in St John's in 1939, if all the family had moved away and his parents had died, maybe he 'boarded' with another local family so that he at least stayed in the place he knew. St John's is small, I spent a fair bit of time there in the summers when I was a little girl, everyone would know everyone else and would have looked out for him so it would be best for him to stay there as long as was possible. Perhaps he became ill or too much for his carer to cope with and rather than 'being put in the workhouse' he was admitted to Harton hospital wing which was near to where you say one of his brothers lived and worked - at least that way someone was near enough to visit him?
Don't discount what JenB told you on the basis of no middle name and incorrect birth year. If he was staying with someone who wasn't family maybe they didn't know he had a middle name and just weren't sure when he was born so gave their best guess. You'd be surprised how many records I have seen with incorrect info, given in good faith at the time, but still incorrect
Boo