Well....
... As Louis learned the hard way, divorcing Eleanor divorces about half the territory of metropolitan France. There's no reason to think that a second divorce / annulment on the same grounds (consanguinity) as the first would not include legitimation of the children and the restoration to Eleanor of her own lands, if only as a matter of stare decisis with the Curial canon lawyers; there is less than no reason to think Eleanor would take it quietly if it didn't. And to effect a dissolution of the marriage, Henry has to get the church onside ... or pronounce a merely secular dissolution which gets him excommunicated and probably hit with a bull of deposition. I mean, if you want to start the Henrician schism six Henries early, feel free, but.... The sons, possibly with John playing both sides against the middle, will side with Mummy, in all likelihood; and the barons can make a good deal of allegedly pious and chivalrous hay with it.
And despite Stalin's question centuries later, the papacy had its divisions (and corps, and armies) then, if only by proxy, through blessing Henry's enemies. Henry was already on thin ice since Becket's martyrdom. And - look, there's a reason for the C of E, and it's rather surprising the Scots didn't do the same early on and that the Irish remained RC: the papacy has a bad history with the British Isles. Alexander II in 1066; Innocent III in 1215, siding with John and denouncing Magna Carta; in Henry's own day and memory, Adrian IV's handing Ireland over to the English; John XXII's anti-Scots policy in the time of Edward and Robert the Bruce.... Alexander III was no friend to Henry, nor was Lucius III, and yet both supported the English over the Irish (I really cannot understand the Irish attachment to the papacy).
In any case, and contrary to a suggestion up-thread, the only way Henry's sons have a shot at inheriting Eleanor's lands, of which Henry had control only jure uxoris, is by taking up her cause, and doubly so in the face of a crooked divorce. So things get very ugly very early.