Marshall may very well be executed for the killing of Richard as part of a deal to patch up relations between the family.
This does bring us to a rather tricky situation however. Henry II had promised to go on Crusade, and Richard as the elder son took up this obligation when his father died. John may be required in some legal sense to do so as well, but I think that given that England has a very shaky dynastic position, this may well be waived to allow John to remain in the country and sort out the succession pronto. After Henry II, the throne passes to his 2 year-old grandson Arthur of Brittany, then John and then we have to go to the female children of Henry II on account on there being no other male heirs (even going back to beyond William I). We actually get Henry V, Count Palatine as a result of this, and he's 16.
Essentially,John needs to marry. John probably doesn't marry Isabel, Countess of Gloucester in this situation (the marriage was forbidden to be consumated by the Pope even when he lifted the forced annulment), and as Isabella of Angouleme is only 1 at this point, she's out the running as well. Perhaps the recently bereaved Countess of Brittany?
John is almost certainly regent in this situation, though we could have a Richard III situation with John (who is suspected of killing Arthur OTL in any case). Unlike Richard I, John has an uncontested hold on the throne/regency as he is Henry's only surviving legitimate son. I doubt this makes him any better an administrator or General, so if he stays in England, he's just as likely to piss of the Barons.