Was the high sparrow withholding his blessing a tell that the faith under him wouldn’t recognize the current monarchy? And after, and even before, the conquest how did the faith work in the seven kingdoms and dorne

No, I think the withholding of the blessing was and is a bid for power within the current monarchy. 

The history of the Faith is a bit vague and confusing – we know that, for a long time, the High Septons were based in Oldtown and both independently powerful but also intently connected with highborn politics (the High Septon of the Revolt of the Faithful was a Hightower, for example). 

So I’m not quite sure how this worked politically. You’d think that the Vale, being the first settlement of the Andals, would have been the historic center of the Faith – can’t imagine they would have been happy with losing the influence and the economic advantages from all the pilgrims. Likewise, how do the Kings of the Rock or the Storm Kings or the Kings of the Rivers and Hills or the Kings of the Mountain and Vale feel about the Faith possibly having pro-Reach bias? My guess is that the Most Devout are apportioned to the Seven Kingdoms in such a way as to ensure that the other kingdoms are represented.  

Dorne’s an odd case – they’re officially of the Seven, but they seem less observant than other kingdoms. How exactly did they relate to a Faith under the thumb of the Targaryens when they weren’t? Not sure.