I'm gonna guess, simply based on an average of hirings, sackings, and finally some retention of good soldiers and the fact you can only employ 250 max, that probably the max number of X-COM troopers might have been up to tripple this max - slightly more if you drag your game year out to the years mentioned in those game texts.
But I would say under a thousand if not slightly more total - this includes washouts since, presumably they were good enough to be recommended to the program and hired before being evaluated then sacked by the Commander, just not good enough to be retained longer than most. Nevertheless I consider these types 'briefed' on the war, and probably hired on by third parties with the security clearance to know of the war prior to it becomming common public knowledge.
They had to have gone somewhere, so that's my speculation for them.
Just how large scale could the first war have been? From the game, we know that four or so bases could do it with about fifty soldiers at any one time if you're 'good'.
The problem is, that makes the first alien war miniscule in scale, which hardly seems rational to me, given the background material of the other games.
How many veterans were left around after the war? According to some fanfics and background material, it might well be absolute bloody loads of them.
The scale of the war was I think miniscule as viewed from X-COM's perspective certainly. However, once the brain mission was accomplished (since I do not know the actual time table between this mission and disclosure to the public - or even whether it was before it of the alien threat), its quite possible more veterans appeared who may not necessarily been solely X-COM operatives.
I would think that any base or alien mop up missions after the brain one could very well have been coordinated with other nations' militaries. I'd like to think that there were several tasks being undertaken that maybe we aren't supposed to be privy to aside from any "X-COM" game book canon.
Just a for instance - what was being done with all that surplus gear you sold? Certainly enough stuff was sold over the course of five or six game years to equip several more squads for several more bases you don't have jurisdiction over.
Another for instance - If enough UFO component material was sold, what might have the scale been of battles in our solar system if any? Presuming plenty of engineering capacity at any parallel outfits akin to X-COM. Even if this is supposedly low to non-existant apart from your Mars sojourn to kill the brain, what about Interceptor or Skyranger technology proliferation if any?
I would think that there would also be plenty of behind the scenes equipment like this in the hands of people just like you doing what they could.
I remember in one of the introductions to the first game, it mentioned Japan's failed attempt to thwart the alien menace prior to X-COM's formation. Its planes and so forth couldn't keep up with any craft and no progress happened so they disbanded their effort- stands to reason any friendlies to the cause would adopt any tools necessary if it knew of the threat facing Earth or even had access to material on the black market for whatever it couldn't manufacture or find for themselves.
Also concievably, blueprint proliferation for the kinds of modules you've made for your bases could have occured at some stage. I sort of view the first war's dynamics this way, that essentially you were not just at the forefront of it but that you were buying any other Earth forces uncompromised by the threat time to build up. An example here is probably the construction of the carrier types talked about slightly before and after TFTD (one was transported out to deep space which I think was part of the backstory for Interceptor).
So these things I think were going on beginning with the first war, with some of these things not appearing until after it was done. I'm pretty positive that (while I'm not sure how well documented it is - it just makes sense to me) after the war was over, there would have been a period of time where there was serious military build up.