C’mon, lighten up – that was* funny*. Funny because it is odd to drink something as specific as milk when you were told not to by a medical professional. If you weren’t allowed to drink any liquid for, say, 24 hours, and you finally gave in and had a glass of water after 18 hours, that would be understandable – but few people crave milk specifically.
Feculant Vomitus, band name!
Serious dairy intolerance here, back when I finally figured out what the hell was wrong with me I had eaten a bowl of raisin bran and about 10-15 min later was pooping…you guessed it, still recognizable raisin bran. I mean that stuff would turn into unrecognizable goo just sitting in a bowl of milk in pretty short order so it must have been riding the express train to make it out the back door still looking like it did.
Great Question!
I don’t have the answer butt I’m like a seagull. 
This specific question has been asked here at least once before; I think the poser was Gonzomax but I don’t recall for sure.
The whole gut transit time is dependent on a number of factors including peristalsis, volume, existing intestinal content and what is being taken in (liquid versus solid, and what type of solids).
The short 
answer is on the order of 15-30 minutes if we are talking about someone eating ordinary foods (not to be confused with an ordinary whole meal) with otherwise normal baseline physiology whose bowels have already been fired up by one of several mechanisms, particularly substances which increase peristalsis. This is not a cold-start time, so to speak–this is an anecdotally-based estimate on my part of what you might see if someone already has an empty gut and a pathologic process acting on it which has maximally engaged their whole-gut transit time potential. That’s where you get those stories of someone with a raging diarrhea who eats corn and has it show up (undigested) 20 minutes later in the stool.
If you wanted to get all Guinness about it, you could get the bowels emptied so they are a nice unobstructed hollow tube, get peristalsis maximally stimulated, and give a liquid oral bolus that perhaps runs through all 25 feet or so in less time.
Chief Pedant and Attack from third dimension: I have personally seen (literally, as an X-ray Tech) persons who came in for a Small-Bowel follow-through for whatever symptom(s) and less than 10 minutes after they started drinking the two or three cups of barium mix have the stuff showing at the far end (rectum).
IIRC, one woman was in bathroom with explosive diarrhea whitish from barium just a few minutes (<10 minutes, for sure, probably around 3-4 minutes) after beginning drinking. Patient was NPO overnight and no special prep prior to exam (like Go-Litely stuff, etc). Special circumstances, of course, and just an example how fast peristalsis can be with pathologies/abnormal conditions. Radiologist and I could hear her belly rumbling over the sound of cooling fans! Almost like a horror movie where things burst from the gut, LOL! When the body wants to expel things from GI tract, it can do so quite readily (as you are saying, of course). I wish it were easier to post the X-rays I have that show this; quite interesting, imho.
On average, though, a normal exam usually takes an hour or two for the barium to get past the cecum (ime). Much longer and radiologists tend to say there is a problem and further testing/imaging is likely warranted if ordering physician desires a more-specific diagnosis. When appendicitis is highly suspected and CT exam ordered, it is routine (at four different hospitals I know of) to wait at least 4 hours post oral-contrast to ensure the mix is within cecum at a minimum - appy’s tend to lock peristalsis down. Gastrograffin is used as barium is firmly contraindicated in suspected bowel perforation, of course. Not preaching to choir, Chief, just my multi-modality imaging experiences over the years.
Thanks; I would not be overly surprised to hear our techs give similar anecdotes albeit not usually formally timed from first sip to first expulsion.
Note for the masses that someone getting a SBFT is almost by nature not normal physiology; otherwise they would not be getting the small bowel follow through. Also, for a lot of people, any sort of medical exam is anxiety-provoking, and among the consequences of that state can be increased peristalsis. So if the pipe is already primed and you put something in the north end, a period of only a few minutes til it rolls out the south end does not seem impossible to me.
I note the interest on this board with rapid transit time while the interest of the broad public (at least as judged by the ads I see) is constipation. Perhaps the next thread will be “What’s the longest you can go without defecating?” Or perhaps it’s already been addressed…