The pertinent AERC rule is 6.2.1.1, which states in part "Riders may
present their equines for the final examination at a time of their
choosing within the one hour period." It is clearly the intent
that the examination itself take place within the hour, but the
rider's responsibility is to have the horse there; the rider
cannot control vet availability. But the rules also state that ride
management may have more stringent requirements (as long as they are
provide to the riders in writing prior to the ride). So, a head vet
can require you to be there in time for the exam to be completed
during the hour. I don't recall ever seeing this done, though. If
in doubt, ask!
Of course, if there should be a line at the final check, and you
stroll up with your horse at exactly 59 minutes and 45 seconds,
don't be surprised if a ride official comes over and checks the
horse's pulse immediately -- just to make sure you aren't trying to
buy some extra time!
Again, this shouldn't really be a problem. The horse should be
recovered well before the hour is up.