@Azel Volantis should have the Banners and any local mercenary/semi-professional border guard free to absorb into the Legions, which should account for that much, yeah. Maybe more technically if you go for pure recruits from former slaves, easily. There's some numbers that can be balanced especially with agricultural reform.
Braavos and Pentos should have plenty of mercenaries that we can entice to join the Legion, as well as the martially inclined youth based groups in the former who make up a lot of Braavos' military during times of war, with the added benefit of plenty of nobles forming that group who know their letters--ready made officers.
I also believe out of the Disputed Lands we can raise one Legion from already-professional forces per Free City, in addition to our 1st Legion.
So with all of those put together, I expect something like 10 Legions as a conservative estimate
on short notice, with 12 being possible if we step up mobilization a notch or two. Definitely 12 Legions if we postpone invasion another month or so, though they'll still have some kinks to work out, they should be fairly organized and disciplined since we're mainly recruiting more seasoned men.