So when that happens you end up in Hospital or Asylum delayed...

Does that mean you get an encounter in that place?

e.g. Movement phase: a vampire messes you up and you end up in hospital. Do you get the encounter next pahase and then the turn after that when you stand up again?

Similarly with the oh-so-rare case of getting arrested... Do you get a Police Station encounter the following turn?

You're not delayed. You just don't have any Encounter once you get there. You go there, get 1 point of Sanity or Stamina, and wait until the following turn. You can still move out on your next Movement Phase; it's just that, usually, most want to hang around a bit to raise it a bit higher.

As for the Police Station...it's worded really badly in the manual, but you aren't really "delayed", you're losing your turn. "Delayed" means you skip your Movement Phase; losing your turn means you skip Upkeep and Encounters as well. What you should do is just put your Investigator Marker in the little Jail Cell space--don't tip it over, just stand it there--and while you're in that "cell", you can do nothing for one full turn. The following turn, move your Investigator into the Police Station boardspace at the beginning of the Movement Phase (after a normal Upkeep Phase), and then act normally. (Jail is very much like being Lost in Time and Space.)

Mylo said:

From the rulebook, page 16:

"The player then immediately moves his investigator to Arkham Asylum. The investigator is restored to 1 Sanity, returning 1 Sanity token to his investigator sheet, and has no further encounters this turn. The investigator may take his next turn as normal."

Note that you are not delayed.