The timer switches players as soon as the card is played, not after the cards effects resolve. This means that when a priest plays a heal to trigger Talented Healer and Altruism and whatever else he may have attached or be holding, the time it takes to go through all of those card animations is taken out of HIS OPPONENTS clock. I seriously think it may be possible to win via time out by abusing this and I just played an opponent who I seriously thought was working towards this goal, until 20 minutes in he actually passed his first turn.
That doesn't seem at all possible at anything other than the slowest animation setting. Seems more likely most players once they get used to what the cards do will have animation set to the fastest speed possible.
The server switch timer as soon as it received the information about which card is played. The card effect (animation) is entirely client-side and there are ways to speed it up. Tbh 20 minute is a plenty of time. Abusing this timer to win is really not a big deal. Deck that draw that much card need to be adjusted eventually.
That's about the last thing I'm worried about when it comes to the triple-priest infinite draw build.