Dear Max: When I was in Public Health back in the fifties,
Australia was claiming over 90%. And they were using a bacterial agent whose name I can't now remember alternately.
I got a chuckle out of stories about sportmen moaning about being deprived of hunting when it got loose in UK. Almost immediately it became clear that the trees were doing better. Gotta see if I can find info about that virus/bacterium pair. It may be too old for Internet.

Edit: the virus of fifty years ago was myxoma virus.
Presumably rabbits must have developed resistance. I couldn't find any mention of the bacterial disease.