This instructional video explains in some detail (It’s long, but interesting stuff) how mechanical computers calculated fire angles in moving battleships. This is essentially the technology Charles Babbage was working on in the 1850s. If he’d managed to develop it, the British Navy would’ve been even more dominant than it already was.
Babbage getting his shit together is of course, one of the main ideas behind the Steampunk genre, with all the gear powered steam driven computers and such.
I imagine in the 1950s mechanical assemblies were still more reliable than electronics for something like this. All those vacuum tubes and transistors probably didn’t get along well with sea water.