While we may never know exactly which of the several candidates for John Gregory's birth is the right one, there's a good deal more we can say about him. Perhaps most significantly, he has a listing in the Royal Navy's earliest volume recording the service of engineers; so far as I know, this entry hasn't been seen or cited by anyone researching his career.
Unfortunately, rather than recording his age, service, and character, the entry simply contains a statement, written across both pages of the ledger: ""This Engineer was recommended by Messrs. Maudslay to serve in the Vessels employed on the Arctic Expedition having been accustomed to locomotive engines his pay to be double of that allowed to 1st class Engineers (Woolwich 6th May 1845) ... Appointed 13th May 1845 Admiralty "Erebus" 1st acting ... Apptd. 1st Class Assistant 6 June 51." So we now know for a fact what we previously only inferred -- that he was recommended by his employers, Maudlay, Sons, and Field; we also know that he was specifically appointed with double wages. I checked the ledger, and a nearly identical statement is written in the record for James Thompson, who served as engineer aboard HMS Terror, but as with Gregory, there are no personal details.
building still stands and is presently the Charlotte Sharman Elementary School. The building reduced Ely Place to more of an alleyway than a street; in 1934, after the land and buildings of the nearby asylum were purchased by Viscount Rothermere (then owner of the Daily Mail), the land on the opposite side of Ely Place was turned into a park named after his mother, Geraldine Mary Harmsworth. At around this time, it seems, the name of the street was changed to match, becoming Geraldine Street -- a name which the UK gazetteer tells me is unique in Britain. This image from Google Earth shows the view looking down Geraldine Street; the school building is on the left and the park is on the other side of the brick wall on the right.
I've already mentioned that John and Hannah's children were accomplished people -- several generations of them worked as engineers, with the exception of grandson Edward John Gregory, who became a noted painter. As to Hannah herself, she seems to have remained in the neighborhood, if not at the same address; in the 1870 census she appears to be living with her in-laws on South Street (modern Greenwich South Street) in Lambeth, but at her her death in 1873 she was apparently resident in the parish of St. Saviour's Southwark, a bit further north and closer to the river. I still hope to locate her grave, and will update this post if I do!