How much time does it take a kidney stone to pass and could this be caused by switching to a vegetarian diet?

It depends: It depends on the size and location of the stone. Smaller stones, and stones located in the distal ureter, have a better chance of passing, and more quickly. It could be a matter of hours, or it could take a week or more. It may not pass, other treatments may be needed. Not sure if vegetarian diet makes a difference.

Varies: The time for passage of a stone is tremendously variable. Vegetarianism is a moral choice rather than a health-promoting one, despite all the hucksterism. You'd do much better to drink plenty of water (never pass dark urine) and comply with other directions depending on the stone's composition.