Rosie> the 1939 has Violet Arrowsmith {crossed out} Cox at 57 Langshaw Road Bolton, f, 19th April 1884 Married Shop Asst Bread & Confectioner with son Jack Arrowsmith born 12 April 1914 Radio Salesman & Repair. So given the marriage to John Cox...... the suggestion is that Thomas is still alive in 1939.......and dies sometime before 1945.

?....given that the 1939 ration cards were effectually changed right up until end of WW2 suggesting the "Cox".
Chempat> I am discounting death year of ancestry tree .........Thomas didn't die in 1929 as Violet states she is still married in 1939.,therefore 1935 death is out at moment... However I am open to the idea that he may have been in Suffolk, BUT, realistically I doubt its him. Now I will show the following which shows Lancs BMD results >
Lancashire Birth indexes for the years: 1882
Surname Forename(s) Mother's Maiden Name Sub-District Registers At Reference
ARROWSMITH Thomas LEATHER Culcheth Wigan & Leigh CUL/36/457
ARROWSMITH Thomas MAKINSON Westleigh Wigan & Leigh WL/30/164
{target]ARROWSMITH Thomas A Salford Greengate Salford GRE/136/93
So I believe the Arrowsmith[Leather} can't be the Suffolk entry......
Births Jun 1882 Arrowsmith Thomas Leigh 8c 262
[target]Births Sep 1882 Arrowsmith Thomas Leigh 8c 250
Births Dec 1882 Arrowsmith Thomas W. Derby 8b 482
Births Dec 1882 Arrowsmith Thomas Arthur Salford 8d 91
So the Suffolk entry does have merit but cannot be proved at this moment...... the Anne Lomax in 1939 could be clue to proving by place of birth?

? Does she possibly come from around Leigh or Bolton area?

Her birth date was 6th April 1884 but in 1939 she is a widow and anc is not showing me enough information.... and there were 17 marriages of an Annie marrying a Lomax in Bolton alone between 1905 until 1939........but after looking at lancs opc I narrowed to four ....grrrrrrrr. Then out of that I would put serious speculation to >
Marriage: 5 May 1906 St Matthew, Bolton, Lancashire, England
Samuel Lomax - 21, Labourer, Bachelor, 22 Topping St.
Elizabeth Ann Collier - 20, Confectioner, Spinster, 22 Topping St.
Groom's Father: John Lomax, Joiner
Bride's Father: William Collier, Machine Painter
Witness: William Collier; Alice Casstles
Married by Banns by: Albert Lewis Curate
with a LOMAX Samuel age65 in 1936 death Halliwell Bolton HALL/1A/488 1936
and my reasoning for this is I have Collier family further back in tree [ Edwards father was married to Collier ]..... but I have only done this from basis of "does Thomas go to Suffolk?"
With some thought here I realistically need to invest in the Arrowsmith/Cox marriage cert or find an image showing if Violet was widowed.......... as Thomas could have long life dying anywhere up to 1982.......? Another brick wall is divorce entries only go up to 1905 on FindMyPast so how does one find records after that year?
Births Dec 1922 Arrowsmith Clifford [ Hodkinson] Bolton 8c 527 is the last born child.
As for information on marriage> my grandfathers 1st marriage happened before [Marriages Mar 1930 Arrowsmith Frank Homer Bolton 8c409] 1939 so the father Thomas is still alive........and second one> Marriages Sep 1943 Arrowsmith Frank- Wilson Lancaster 8e1579 was an RO one because he divorced. {Noting both Frank & Stella deceased, as with 1st wife and child}