The Le Puy could be busy in May, certainly with French walkers who come out in droves, and walk the first two weeks to Cahours. My wife and I walked the route on our way to Lourdes last May and June. We found we needed to book ahead for a couple of beds, especially in the towns and villages where there was only one place to stay. So you might find yourself competing for a bed here or there (it will be wise to call ahead a book). It is by far the most popular French pilgrim trail as compared to the other trails in France. I think some 20,000 people walk it, mostly the first half at least. And mostly French walkers and few foreigners.
The Arles route is quieter and far less people walk this route by probably a factor of 1/10 as compared to the Le Puy route. So for most of the time you have the trail to yourself. However pilgrim style accommodation is further apart, and many of the stages are longer, so you need to plan for this. Given the distances and lack of pilgrim style accommodation is some stages, again you might find you need to call ahead, though I sure you will find a place most nights without too much effort.
The Vezelay is the least busy route of the three. I have not walked it but all accounts I get the impression, you have the place to yourself.