How long does caulk need to dry in a 1" gap?
safe to use the shower is a relative question.  when will it be safe to use it so that its use doesn't compromise the caulking seal?  never - it was already compromised the moment you did it that way.

caulking shrinks as it dries, and the speed at which it dries changes how it polymerizes.  when you make a huge blob of caulking, you seal in much of the solvent and that changes how the sealant cures dramatically.  essentially, the inside of the caulking is still wet with solvent, which prevents the material from properly linking together to make a waterproof plastic membrane.  since that waterproof membrane has to adhere to the surrounding surfaces to keep water out, it can't.  thats the key to success.  

too much caulk means the membrane never fully forms.  you get a layer on the outside just thick enough to hold in the solvent.  now, when you wet it with water, the water also seals in the solvent.  since there is flex in every joint, everytime you stand on it or near it, or swing the door, or fart strongly, the joint moves.  however, instead of a strong, well adhered plastic membrane flexing and returning, all you are doing is tearing the thin outer membrane and letting in a little air to further cure a tiny bit inside the tear.  this will continue for weeks until the boundary between the new sealant and the substrate is completely sheared through, and even then you may not have fully cured caulk.  as the curing process continues, the caulk will shrink until that boundary is not only fully disconnected from the substrate, but also shrunk to the point where the boundary is a gap.  then water gets in again.

to do it right, you have to follow these steps:

1)  remove all old sealant mechanically.
2)  using whatever solvents are necessary, clean off all remaining traces of the preexisting sealants (this is important, as you need new sealants to bond to the substrate, not the old sealant)
3)  mechanically abrade (sand) the new joint areas to promote proper adhesion.
4)  if the gap is bigger than 1/4", fill it with a compatible seam filling rope or extrusion that is compatible with your sealant and the type of movement you are going to have.
5)  seal with a good quality urethane or silicone sealant.  don't use latex sealants inside showers or bathtubs, no matter how much it says you can on the packaging.

and for the love of god, do your work carefully and with some attention to craftsmanship.  it looks like you put that sealant in and smoothed it out with an old shoe.