I think the countries that will be pro axis/pro allies neutrals will be countries that are typically active in AA games (belonging to one side or the other), but have not officially joined the war by the games start date (around June 10th 1940). If they are leaning to your side you don’t have to invade (fight) you just move units in and take ownership (probably in the combat phase).
If they are hostile to your side you will have to attack the units printed on the map. New countries that have been added to AA40 expansion of say The Middle East or Africa that had not yet joined the war should also follow this format.
There could be something like move into a pro your side in combat move and take ownership the IPC’s are yours, where if you come in non combat the tt remains pro your side (you only occupy it) and don’t get the IPC’s. There may be some reason that you may only want to occupy a tt (avoid some kind of consequence, like others turn on you).
There will be certain countries that have been traditionally neutral like Turkey, Saudi, Spain, Sweden, maybe Switz, parts of S America etc. Either side will have to fight there way in like you do with Mongolia for these true neutrals. I could see certain countries having alliances, that if they are attacked others will join the other side like in S America. I hope there is no link between say Spain and Turkey, and if there is you can avoid it with political pressure in the individual regions. Like say you have Turkey surrounded, and attack them it keeps Spain out. I would not like a rule that says attack one true neutral and all the others join your enemy (but it could happen). If Germany/Italy attack Turkey to open up the Black sea (to invade the Caucasus) allows the US to start putting troops on Spain that would not be cool. The axis would have to invade both at the same time to watch their back. If an attack on Turkey brought the Saudi’s to the other side or vise versa, that I could live with, at least they’re in the same region. I don’t know if they actually had a political alliance though.
Maybe NO’s will dictate if/when you can invade a true neutral (w/o consequences). If you have say a Middle East NO (control of Persia, Iraq, Syria, Jordan and Egypt) you can invade Saudi or Turkey (Turkey may also include another NO w/Balkans making it more difficult). If you achieve an NO with the Scandinavian tt you can now invade Sweden Etc. Control the region (NO) gain access to true neutrals. I know its a stretch, and NO’s most likely won’t have any effect on neutrals, but it was worth mentioning.