Will botulism growing in my home-canned vegetables pop the lid?
One reason botulism is so scary is that you can's see, smell, or taste the bacteria growing in there. The only way to know for sure if botulism is growing in food is to have it tested by a laboratory. 

Fortunately, though, you can see its evidence. 

Yes, a puffed-up can or a jar w/ the lid popped up means something's growing in there. Discard the material; do not consume it. 

Discard a metal can if there's any visible opening, no matter how small, if the ends are bulging, if the seam doesn't look intact, or if there's a leak.

Discard a jar if the jar appears cracked at all, if the pop-top doesn't pop when opened (meaning there's no longer a vacuum inside), or if the seal appears damaged.

As for how long it takes for a problem to arise, I'm not sure; I've had the same question.