Do you had a picture of the movement?
I have never seen an issued watch that looks exactly like that before.
The dial is similar to the O&W watches that took their design influence from the UK MOD Def stan for watches. The lack of issue markings on the case back, the use of spring bars over fixed bars means I think this is homage watch and not an actual issued watch.
With regard to the movement. The only military issued Rolex watches were the well know various models of submariner and the lesser known HS10 Explorer. (Ref 6159) The latter was issued in tiny numbers.