My guess is the late 1970s. I had a Craftsman socket set in a plastic case that I bought around 1980.
I have a Husky socket set from around 1980 that came in a plastic case. And a more recent SK set that still came in metal. I recall having Makita and Porter Cable tools into the 1990s that still had metal cases.
So I can't say that plastic cases took over in any one year, or even decade.
Plastic, in one way or another, dates back a LONG time. The 19th century saw widespread use of hard rubbers (gutta percha) and celluloids, to name but a few. Bakelite hit the scene in 1909. Styrenes, acrylics (perspex/lucite/plexiglas), nylon, polyethylene (the thermoplastic responsible for blow molded cases) and teflon were all in use before WWII. Polypropylene came shortly later.