No...not in the way that she's suggesting. The very first thing you need to know about vegans is that they lie. Question every claim they make and refuse to take PETA, PCRM, etc, as a reference.
Depending on where in the world that you live, slaughterhouses and dairy farms may shut down, but here in the US, meat production is up. Major slaughterhouses are owned and operated by major meat companies, like Tyson. They're not closing down.
Dairy production ebbs and flows. Here in the US, too much milk being produced....mostly because govt subsidies encourage dairy farmers to produce more and more milk. As prices drop, farmers will sell some cows for beef, cut back on production until prices go up again. Then they'll expand their herds again until they're producing too much milk.... again....and prices will go down. It's a never ending cycle. Some smaller dairy farms get tired of it and sell to larger farms that are better able to ride out the ebb and flow.
No, the world is NOT going veg*n. A poll last year showed there are fewer veg*ns than 8 years ago in the US.
Besides, actual grown up research shows the vegan diet can't feed the planet's growing population. See link.
(Veg*n is shorthand for vegetarian and/or vegan)