The M-14 was fairly slow to be issued. There are photos of US Army soldiers in Berlin in 1961, during the Berlin Wall crisis, still armed with Garands. I’s suggest it would be extremely unlikely that it was rolled out any faster to the USN. Maybe by the mid-sixties at which point it was widespread in the Army and Marines.
The Thompson had been officially declared obsolete in 1957 but was still a limited standard item with the USN. There are plenty of photos showing Thompsons in use in Vietnam. Some of those, particularly later in the peace are likely to be private purchase or barter items but if you are interested in the late ‘50s I’d say that fielding a Thompson would be quite OK.