[Cairn]

I know that at some point along the way, it must be impossible to continue doing missions for all three. However, I've been given the mission to join the militia, the mission for joining the Merc's, and I've brought the lamb and 1000 gold to the Monastery and I've become a novice.

I've looked at my failed missions, and there is no indication that I can't still accomplish those missions to become a merc and a militiaman.

*Minor Spoiler*
(But then I completely forgot about the deadline for accomplishing the Thieves Guild mission, and that mission is not in my failed mission section either. Maybe it's waiting for me to actually kill the mission-giver before it gives me the failed mission note.)
*End of Spoiler*

Anyhow, my question is,
At what point in each of the three factions can I no longer join either of the other two factions? For example, with Fire Mages, is it when I decide to actually be a Fire Mage?

[stratos5]

As soon as you become a Novice, you can no longer join the other factions. It is possible to do nearly all PreQuests before joining any faction, but once you decide for one, you cant switch.

So its not possible.

[Dino]

Short answer: No
Long answer: No