Does fast one-day prototype manufacturing exist (with components, etching and pick and place under one roof)? What is the fastest possible?
I'm not aware of any, but it's not something I'd expect to find. 

Assuming you have a complete design ready to go, living in the continental US or Canada you can get parts overnight if the quantities aren't huge.

The fastest turnaround I've ever seen as a standard offering from a board house is 24 hours, and it's not cheap. That could also show up next day, although you need to place the order early in the day and coordinate it with the board house in many cases. You can look up individual board houses for their rapid turn rates, but they can add an extra 0 on the end of the price depending on what you need. They might not be able to do it if your board is complicated enough, or they're already at capacity. 

For rapid prototyping you could mill or etch your own PCBs, but that's a non-trivial thing to do for any sort of compact or complex design though. 

Depending on volume assembly houses could do everything in a day, but again you usually need to coordinate it with them in advance so they can fit it into their build schedule. 

If you really needed something fast, and coordinated it well between your vendors and assembly house you could have everything ordered on Day 1 and a built unit in hand by the end of Day 2. Provided nothing goes wrong and all your vendors could actually meet the demand.

The price premiums attached to this however are incredibly high, digikey charges something like 60 dollars in the US for overnight shipping. That's nothing compared to the rapid rates for board and assembly houses. I've done it for the odd project, but those were in instances where there was already a time crunch and money was not a huge concern.

From my professional experience however it's just not worth the hassle. It also is more probable you'll mess something up if you're rushing. Plan your project out so you can do other useful work while waiting for your stuff to show up.