It all depends on how you define "ASAP".
I've seen books take two years or more from signing to publication, when not a moment was wasted in their publishing schedules. That was, for them, "ASAP".
I've seen books get hustled through the publishing schedule a bit more quickly than they would normally be moved, which have come out within four or six months of being signed.
I've seen books written, edited, printed and published in under a month (for example, when Michael Jackson died numerous new books about him popped up in a couple of weeks).
It all depends on how topical the book is and how much extra money a publisher is happy to throw at it to get it out on the shelves as quickly as possible.