If you want one dated 75-80, then like you, I reckon it's a Hamilton or CWC W10 Mechanical General Service Watch.
https://cwcaddict.com/general-service-mech
Maybe start with a WTB here or on MWR.
Smiths (and probably 1945 WWWs) would still have been around although not marked in your date range.
I did have a Hamilton W10 ten years or so ago. Sold it for around £175, so just a picture of my Smiths.