I'll drop my own estimates as well, for fun.
Personally, on the very scarce sources of population for Thedas I'd guess the population to something like this:
Tevinter: 10
Orlais: 10
Free Marches: 15
Nevarra: 5
Antiva: 5
Qunari: 5
Anderfells: 2 (thought it was around 5 before, but after reading around a bit I don't think so anymore
)
Rivain: 1
Ferelden: 1
Dalish: 0,5
Total: 54,5 million.
Orlais is huge after all, and their heartlands are very densely populated (I imagine Tuscany during the middle ages personally), and even though much of the area is scarcely populated it is massive.
Tevinter lies in a lush region, Minrathous has a population alone of around 1 million, and the oldest settlements in Thedas should be here.
The Free Marches were described as the breadbasket of Thedas, and when they unite their forces even the strongest nations tremble. I imagine them much like the scattered city states of Germany and Northern Italy.
Tevinter:
I actually think Tevinter has the largest standing army, perhaps something along the lines of 100.000. Their climate should allow for a sufficient surplus of food. There are few other professions for those who are not slaves, and not wealthy merchants (Krem is an example), so I expect this is where most of Tevinter's "middle class" is employed, as well as their not so well off mages. Add to this a number of slaves kept for the express purpose of war, although I think most such slaves are guards to wealthy mages and magisters, and not outright soldiers.
Tevinter is after all in constant war with the Qunari, and thus needs a large standing army.
Note: Tevinter probably also keeps a considerable navy in order to safely ferry their attacking armies to Seheron every autumn. But most of the ships are probably merchant vessels which only serve in war for periods of time.
Standing army: ca 100.000
Qunari:
Their army is probably the strongest in terms of actual fighting ability. They have a discipline and skill, as well as just pure physical advantages that give them a decisive edge. However, their numbers are probably not the greatest. Only those "meant" for military service serve in the Antaam, and the Qunari soldiers are full time soldiers, living for nothing else. I'd say there are roughly 50.000 all together.
Standing army: ca 50.000
Orlais:
Now this is frequently established as the primary power in Thedas. But their military prowess the last years seem lacking at best. Both Nevarra and Ferelden have bested them in war, although not easily.
Personally I imagine Orlais has the problem of quantity over quality. Though their chevaliers probably possess both skill and discipline most of Orlais' wartime strength is probably poorly armed peasant levies that the various lords and ladies of Orlais contribute to the army. The Royal Army itself is probably not huge in peacetime, the expenses related to "The Game" probably far outstrip those afforded the army (and navy). I'd say Orlais probably only keeps about 30.000 outside war, a number which can be boosted to around 100.000 in wartime.
Standing army: ca 30.000
Maximum size: ca 100.000
Free Marches:
Each individual city of town probably only keeps a small professional guard force, similar to Kirkwall's, and then trust mercenary forces in their internal rivalries. I doubt there are more than 10.000 professional soldiers employed in the Free Marches outside instances when they are engaged in war against an outside foe. In such times conscripts and mercenaries probably allow them to rival any power.
Standing army: ca 10.000
Maximum size: ca 100.000
Nevarra:
They can beat Orlais, but only when most of the Orleasian army was engaged in Ferelden. They're probably more militarized than Orlais, they seem to put more stock in martial prowess anyway. I'd say they have roughly 40.000 men under arms, they could probably increase that in wartime though.
Standing army: 40.000
Anderfells:
They're "the most feared" country in Thedas. I think the army they have is fierce, but I do not think it large. The king's power is mostly limited to the capital city apparently. So most of the forces are probably local militias and noble retainers. I doubt the king commands more than 5000 men, ish. Add in the men the nobility keep on guard against darkspawn and I guess you have around 15.000
Standing army 5 - 15.000
Antiva:
Is said to maintain no standing army. But they must keep some forces none the less, if only for ceremonial purposes, and a bit of city guarding. I'd say maybe 5000. They're essentially a larger version of a Free Marcher state, and probably has a pool of mercenaries to draw from in a crisis (or the merchant princes do at least).
Standing army: ca 5.000
Rivain: A small country with peaceful relations to their neighbors, easily defensible across their land bridge and largely lawless. I'd say they keep around 5000 soldiers as well.
Standing army: ca 5.000.
Ferelden:
In peacetime the Ferelden army is probably not very large. The economy of Ferelden would not permit it, nor its lords' fondness for independence. But Ferelden has a tradition for martial skill, and most healthy adults seem to be at least proficient with a weapon, so I do not think it unlikely that the number of men and women that could be easily conscripted is rather large. The nobles all keep their own forces in peacetime, while the king retains a small Royal Army of a few thousand to guard Fort Drakon and the Royal Palace, as well as the border crossing with Orlais. But in war I think Ferelden can field many times what we saw at Ostagar. After all, even after that disastrous defeat, and a civil war we still had a decent army to assault Denerim with.
Standing army: ca 5.000
Maximum size: ca 25.000
Dalish:
The Dalish are quite tricky. I believe their number to be around 500.000. Thedas is sparsely populated, there are lots of places the Dalish can go. They've been around for 700 years, and their numbers should logically not be in decline (what with having magical healing available, and being able to trade when food is scarce). But that's just guesswork. Anyhow, we do see that a hastily assembled Dalish army in Origins is equal to their task and quite formidable in its own right. Although they do not keep a standing army most Dalish are skilled fighters, they have an abundance of mages, and at the very least it seem to me that in Ferelden alone there are enough to construct a small army in the space of months. The Dalish could probably put together an impressive army if need be. Although they'd resemble a horde more than an actual army in my view.
Others:
Wardens: ca 3000 I think. More than a thousand in the Anderfells, several hundred in every other nation but Ferelden.
Templars: 15.000 seems about fair, yeah.
Circle: I doubt there are so many mages. That would mean each circle has/had more than 1.000 mages, and I doubt that. I think there are probably only a few hundred in each. With 15 circles I'd say there are about 4 - 5.000 circle mages all together, with about an equal number of apostates and hedge mages about.
Inquisition: I don't know. At the end of the game we might have around 10 - 20.000 soldiers. But I doubt that will last.