Found out some info on the Longines-Wittnhauer but still don't know the model.
Here are some information regarding the watch with consecutive number 13.375.961 : The movement, dial and hands were put together in Switzerland. The gold watch case and the gold bracelet were produced in the USA The watch was sold in New York on February 24th 1966.
The Sheffield also looks like a 1960s type of watch to me. These are classic dress designs so it's difficult to date them without a clear photo of the movement and even then it will be hard to get an exact date. You are fortunate with the Longines-Wittnauer.
Sheffield watches seem to have been low to low-mid grade Swiss made watches, I would guess for the UK market. The various examples I've seen recently on ebay seem to be pretty generic. I doubt if you will ever turn up any info on this watch apart from possibly seeing it in an advertisement of that period.
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