Before I get going, I warn you that this will be lengthy.
I studied long and hard the few Atkinsons living in West Bromwich, followed them forward and accounted for all but one who was indeed a Thomas Atkinson. He appeared to vanish after the 1901 census whilst in 1903 your Thomas joined the Royal Navy. So, we had a Thomas disappearing after 1901 and your Thomas apparently appearing in 1903 aged 20 with no earlier record of him.
When a likely death in 1962 Bodmin for a Thomas bc.1876 was mentioned by osprey it 'fitted' with the Thomas I'd found who appeared to vanish - and now trish1120 has given burial details which confirms your Thomas did indeed die in 1962 and thus some years older than you'd thought.
I note that on his navy record his birthdate is 1/8/1883 (in 1939 year has changed to 1885!!). However his birthday of 1st August is consistent on both. If it's any consolation he appears to have inherited the changing birth year gene from his father who was named Charles, not Thomas.
I first find Charles Atkinson appearing in West Bromwich in 1861 - he is shown as aged 33 (no status given), a Field Labourer, born Rugeley, Staffordshire - he is a lodger.
In 1871 still in West Bromwich - now 45, again a lodger, unmarried, Millman Ironworks and birthplace now Lichfield, Staffs.
By 1881 - still in West Bromwich - transcribed as 'Atkiss' on Ancestry - with wife/children:
Charles Atkiss 51 Iron Worker born Lichfield, Worcestershire
Mary Jane " 33 born Oldbury, Worcestershire
Sarah 10
Charles 8
Eliza 7
Thomas 5
Josiah 2
John 2mos.
Looked for a corresponding marriage for Charles and Mary Jane but couldn't find one in the right time frame. Eventually discovered that they didn't marry until 1885:
Charles Atkinson 48, bachelor, son of Joseph Atkinson (deceased), labourer married Mary Jane Jenks 39, dau. of Thomas Jenks, Wire Drawer on 14/12/1885 St. Johns, West Bromwich - address for both given as 81 Sam's Lane, West Bromwich - in 1881 they were living at 18 Sam's Lane!!
Checked birth entries for their children as shown in 1881 to see if registered as Atkinson or Jenks - from Eliza through to John they were all registered as Atkinson. All 6 children are shown as born in West Bromwich - the elder 2 (Sarah bc.1871 and Charles bc.1873) not registered as Atkinson, nor in their mothers maiden name of Jenks. The closest birth entries I could find were a Sarah Atkin in Dec.qtr.1871 and a Charles Atkins in Sept.qtr.1872 but no idea if these are them.
By 1891 Charles is 60, Mary J. is 43 and they are now with just 3 children - Thomas 15 and Josiah 12 plus a new daughter May who was born in 1889.
Charles Atkinson's death recorded Dec.qtr.1895 West Bromwich as aged 60. So, as you can see from the various records his own year of birth varies between 1826 and 1835!
1901 - still living in Sams Lane, West Bromwich is Mary J. Atkinson, 52, widow with son Thomas 24, bricklayers labourer, dau. Eliza 27, son Josiah 22 and dau. May 11.
Thomas Atkinson, son of Charles and Mary Jane, had his birth registered in Sept.qtr.1875 West Bromwich and appeared to vanish after the 1901 census. Then we have your Thomas joining the navy in 1903 giving a birthdate of 1/8/1883 who didn't appear to exist before then.
I truly think these 2 Thomas' are one and the same - however, I cannot explain why when enlisting in 1903 he states he was born 1/8/1883 aged 20 when he was in fact 28, nor why he gave his father as Thomas when he was in fact Charles. He was 20 when his father died so no excuse for forgetting his name.
As I said at the outset, explaining all this makes for a lengthy post. So, the only thing I can think of to try and confirm any of this is to get the birth certificate of Thomas Atkinson reg'd Sept.qtr.1875 West Bromwich in the hope his birthdate will be given as 1st August which is the birthdate he gave in 1903 when joining the navy and again on the 1939 Register.
Also found out a bit about Mary Jane Jenks but will leave you to digest and try to understand the above before mentioning that.
Annette