The answer: it really depends! There are a lot of factors:
(1) Plebes report time: the later the day, the possibility the longer the wait lines
(2) Medical issues: if something is discovered, the Doc might take a look at the candidate; also the possibility of how many immunizations one needs and line
(3) Haircuts: once again the longer the line, the longer it takes to get in the seats (while USNA barbers are great, they aren't the USMC ones who do haircuts in about 20 secs each)
(4) The ability to even recognize a son (daughters might be easier) due to haircut; this isn't to mention that they will be flying past you (literally you have about 10-15 secs from the time they are released in the back of Alumni to the waiting bus).
I'd say in the best case scenario, the minimum time is about one hour. However, it could be as much as 2-3.
You could always "welcome" your new plebe at the mid-store where they are dropped off after they leave Alumni; you normally can get more than 10-15 secs of time, since they fall into platoon formation.