Quick question. If we all went vegetarian, what would happen to the livestock?
Interesting question. My educated guess is that there will always be a demand for cow's milk; it's not as if human babies who cannot be breast-fed can really feed on much else. A milk industry will inevitably lead to the existence of male calves, which might as well be slaughtered for food. Same with the chickens and the eggs, if you'll pardon the expression - I can't see eggs disappearing because they are used in medicines and all sorts.
And then in the UK, one of our bigger carbon sinks is grassland and livestock are a fairly integral part of the management of said grassland.
Then you've got wild animals which still make up part of the food chain. Deer and so on are still going to have to be culled from time to time if we want to keep bio-systems balanced.
Finally, there are plenty who just will not give up meat. The likes of Bill Gates who, when not flying hundreds of people thousands of miles in private jets to birthday parties and then lecturing us about the environment, has stated that he is not willing to give up his daily hamburger. I know quite a few South Americans, very well-educated and well-travelled ones who regard it as their birthright to consume as much beef as they want.
Honestly, I can't see the world becoming wholly vegetarian, at least not for a while. What I suspect will happen is that in more enlightened parts of the world meat consumption will form an ever-smaller part of a 'flexitarian' diet, there will be trade protectionism for producers, and prices will rise sharply to reflect higher expectations on welfare, regulation and reduced economies of scale. And, almost certainly, punitive taxation.
Elsewhere rainforests will still be slashed and burned for cheap grain-fed meat with little thought for the consequences.