Great POD.
John would only be about 20 and not terribly experienced. John had shifted allegiances once he felt that Richard was going to win and once Richard is killed he will be back begging forgiveness from Henry very shortly.
Then Henry dies and John comes off as a bit weak, young but obviously the most clear-cut candidate. He wasn't marred by the scandals of having tried to rebel against a sitting king as much as he, in fact, was after Richard's death OTL, so that'll probably soon be forgotten, just a bleep on the radar compared to what Richard and The Young King has done.
Now, 3-year old Arthur of Brittany can and will probably be used as a counter, as mentioned above, but once John frees his mum (Eleanor) she will do exactly the same she did for Richard and move heaven and earth to secure backing for the new king, and her last surviving male child.
Marshal would support the sitting regime.
I feel pretty confident given her record OTL that as long as Eleanor is alive, and Marshal back in the fold, John will be king, and pretty quickly at that. Coronation July 1189 or thereabouts.
I'm not so sure John would go on crusade, though, despite the threats to the (rump) Holy Land. I think he'd find all sorts of excuses. And Phillip A is probably to clever to go anywhere without John out of the way, so he stays and finds some excuse of starting to chip away at the Angevin realm. A younger and less experienced John wouldn't likely do much better, probably worse, than John OTL in keeping the French lands.
The economy might be better because Richard's ransom stays in the coffers but gets used for more wars instead. If John doesn't get killed in action, expect him to be pretty much out of the continent by 1200. Aquitaine might even fall, too, once Eleanor is gone. If the 'drive' in John, or what you call it, is there for him to behave more or less OTL-ish, we would have Baron's War-conditions in England in the early 1200's if not before.
Louis VIII is too young to really take advantage of it at that time, but perhaps Phillip Augustus would? I see either a French invasion in the 1200's or England split up between warring factions for the 1200-1210's duration, which allows Phillip to quietly take the rest of Aquitaine without too much interference.
If Arthur survives/is removed from John's influence early on and John is killed, either in the near-certain war against Phillip A, or in the Baronial Wars sometime before 1210, then Arthur could be the man back in England at that time to take control, probably helped by Marshal. It might be a replay of Louis' intervention OTL but with 'the real king' returning (i.e. of the Plantagenet-line), just like in the myths. Well, kind of
With Arthur's family in charge for the foreseeable future, how would their policies differ towards France? That would be the big question. But even if you have an unstable, de facto leaderless England for a while in the early 1200s or if you have Arthur's line in charge, you must certainly have butterflied away the Hundred Year's War, and that would mean Major Changes down the road ...
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Edit: I don't recall the timing of John's shifting of allegiance in the war between H2 and Richard. Was it before or after the unhorsing? Maybe it doesn't matter that much.
More importantly is probably what Eleanor is going to do about Marshal when she learns he has killed her 'favorite' son, even if in legitimate battle. She might push John to have him executed to make an example, but I'm not so sure. Eleanor was pragmatic if nothing else. She might have wanted Marshal sidelined for some years, until she was too old to influence much and John had too great a need of Marshal's experience to resist calling him back to service.
Not that I would to hinge everything on one person, but it'd be a major loss to the war-effort if Marshal is sent to the axe, or just exiled in the Angevin backwaters somewhere, while John and Phillip duke it out in the early 1190s.
