I'm not an economist, but global economic collapse seems unlikely. I've assembled a small collection of quotations here from different sources with hypotheses about the economic impact of a big shift toward vegetarian/flexitarian diets.
What would happen if the world suddenly went vegetarian? (BBC Future)
People formerly engaged in the livestock industry would also need assistance transitioning to a new career [...] Should we fail to provide clear career alternatives and subsidies for former livestock-related employees, meanwhile, we would probably face significant unemployment and social upheaval – especially in rural communities with close ties to the industry.
The dark side of plant-based food – it’s more about money than you may think
Consider the launch of a whole new range of laboratory created “fake meats” (fake dairy, fake eggs) in the US and Europe, oft celebrated for aiding the rise of the vegan movement. Such trends entrench the shift of political power away from traditional farms and local markets towards biotech companies and multinationals.
Vegetarianism is good for the economy too
The additional food that would be produced as a result of a shift to a vegan diet in the US alone could feed 350 million additional people. [...] The value of this food surplus would also offset the loss from the decrease in livestock production. Economic studies show that animal agriculture in a majority of western economies accounts for less than 2% GDP. Some studies in the US suggest a potential reduction in GDP of about 1% but this would be offset by growth in other plant-based markets.
The Economic Case for Worldwide Vegetarianism
Out of all the world’s countries, the U.S. would save the most by curbing its taste for meat. Due to its very high per-capita health-care costs, the country could save $180 billion if the population ate according to recommended guidelines, and $250 billion if it eschewed animal food products altogether—more than China, or all of the EU countries combined.