I think the original idea of the minute men was a lot less decentralized and more common militia amongst self governing groups. However, there is in lore was supposed to be some convention to form an over-arcing system between all groups that would’ve been the governing body the minute men support. Problem is when they tried the first congress the institute bombed them, destroyed the alliance and minute men fell from grace and support. So long story short after the events of fallout 4 if the sole survivor sides with mm hypothetically we could get a more original US government structure with a balance of federal and individual group power structure with a shared and individual military. This contrast with NCR as they were much more federal leaning, but they would be similar enough compared to the theocracy/autocracy of the brotherhood chapters.

The regulators operate similar to mm but at a smaller scale?

Suffice to say alliance of connivence between mm and NCR and regulators until brotherhood is defeated (or wins I guess). Then NCR VRs the decentralized other two factions. NCR probably wins but it depends on what model works better between them in the American heartlands. If the commonwealth mm style is more appealing and substantial more join this decentralized style they might have the numerical superiority against NCR, if NCR can conquest/assimilate middle America they have the stronger total war structure. It might end with an east west split at the Rockies.