Thanks Jim.
I think the movement is calibre 37.5.
Dating from movement numbers (because they were produced in batches) is difficult - but comparing it with my own, I would guess it dates 1938, probably with a case number 943****
I think the spot by
@JimInOz (showing ‘Catalog picture from collection of
@Tire-comedon’) is on the nail (good spot Jim!). There is another image of a CK1055 on Page 98 of AJTT. I show an image which compares the OP’s watch with another CK1055.
At this time, Omega made pocket watches with calibres 37.5, 37.6 and 38.5, with many variations of dial, hands, stem/bow, case and case material. What is unusual here are the 1/5th marks on the minute scale - which would normally have been used on a watch with a centre second hand (such as a chronograph). The reference 1055 (CK1055 for steel, OJ- /OT1055for gold) had these 1/5th marks - and also the (lovely!) pontife hands.
Whilst it sounds plausible that this watch was part of a travel case, Omega didn’t use this calibre in a travel case (or other ‘unusual’ forms). The sharp edges to the case were not because of it fitting into a mount or stand - they were the style of the case. It is most likely that the original stem/bow were at some time removed - and if the OP shows us a detailed image of the top of the case, this might be apparent.
In summary, my bet is a CK1055 dating about 1938 with ‘modified’ (removed) stem/bow.
As for servicing, if the OP so desires, it shouldn’t be too difficult a task for any (proper) watchmaker.