I've had a look around. Nothing conclusive, but:

It's certainly stated in a number of places that the mbira was banned in pre-Independence Zimbabwe; this site adds that playing it was an imprisonable offence, but also that the "colonial government feared its magical powers", which rather weakens credibility. No dates or references are given.

Chartwell Dutiro makes no mention of any formal proscription, but says that Salvation Army missionaries banned it in some places.

AFRICANEWS states "The Rhodesian Government, fearing that mbira music would inspire revolution, banned it from radio". This seems altogether more likely than some sort of official attempt at governmental level to ban the instrument altogether, though we appear to be talking about the "UDI" administration of Ian Smith, which I suppose might actually have been stupid enough to try.

It looks as if it was mainly a case of missionary interference again, with a recording and broadcast ban imposed from some time in the 1960s or 70s (presumably during the breakaway administration led by Smith), perhaps lasting until the fall of that government. Of course, I may be quite wrong; perhaps someone could give us precise dates and details of any legislation?