If Richard is killed in a skirmish in Cyprus or has the same sort of fate as in OTL, ie shot by a crossbow bolt during the siege of some castle or town, then presumably John has no real challenger as his late elder brother Geoffrey (k 1186)'s posthumous son, Arthur, Duke of Brittany, is only five - but Richard named Arthur as his heir in a recent treaty of alliance with King Tancred of Sicily which he signed after a mini-war with him en route to the Crusade so A does have a legal claim once he is old enough to fight. His mother Duchess Constance, the ruler of Brittany (Geoffrey was only her consort legally, though in practice he ruled with her as insisted on by his father Henry II when he forced Constance to marry him), and her - soon estranged - second husband, Earl Ranulf of Chester, will be defending A's rights and may go to the French royal court to lodge a claim once King Philip Augustus gets back from the Crusade. But if PA did name Arthur as legal heir to England, Normandy, and Anjou very few if any barons would be willing to fight for a child rather than John who by this time would have secured the crown of England and the resources of E, Normandy, Anjou - and Aquitaine too provided that his mother Eleanor , its Duchess, backs him against Arthur as better able to ensure stability. (All backed John in the succession dispute in April 1199 when Richard died in OTL though Arthur was by then 12 and so arguably old enough to rule with help.)
At best Philip, who had constantly backed dissident Angevin relatives against the then King of England since his accession in 1180 (Henry the Young King and Richard against Henry II and then John against Richard) would receive a disgruntled Arthur's envoys at his court as overlord of Anjou and Normandy and either try to blackmail John into giving him important border areas of Normandy in return for recognition of his claim (he grabbed the Vexin from Richard in 1193 when he was in prison in Germany in OTL) or recognise A but not be able to do anything about it. A border war would follow until a truce was patched up - and once A was adult, Philip would use him to start a new civil war in Anjou and Normandy.
What he could grab would depend on how many nobles the irrepressible, sexually voracious, tax-greedy John had alienated by then - John seems to have been unable to help himself in his sporadic predations and lack of an ability to keep his elite happy long term as he kept on making unnecessary enemies in OTL, though if the crown had fallen into his lap in 1192 he would not have had the bad reputation he did in OTL for treachery to Richard in 1193-4 or have lost much of his lands in France to Philip and so needed to raise a new army by extortionate taxation as he did after 1204 in OTL. (He already had a dubious reputation, though - for deserting his dying father to seek Richard's goodwill in 1189 and for messing up his first major role in his expedition to Ireland in 1185, neglecting military and political 'work' there and insulting the locals.) And if a predatory or arrogant John had made enough enemies to lose parts or all of Anjou or Normandy to Philip and Arthur in an attack c. 1200-1205 (A could attack from the West in Brittany so J would have to fight on 2 fronts) then he would be short of cash and military strength and raising this by harsh extortion as in the OTL later 1200s. He might still end up facing a Magna Carta revolt unless he had managed to capture Arthur by his occasionally brilliant generalship - and if he did he was capable of killing him despite the repercussions as he did in real life. John possibly killed A in a rage , but other killings of his - eg starving the ex-rebel De Braoses to death in prison in 1210 - were calculated spite and terror, and so he was likely to end up with fearful enemies deciding to strike first.
Other thoughts - if there is no Richard to put Henry of Champagne (his half-sister's son) on the throne of the Kingdom of Jerusalem in 1192 when Conrad was killed by the Assassins - I do not think it likely that R was responsible for hiring the killers, though Philip claimed this, and Conrad had local enemies too in Syria - then Henry is unlikely to get the throne. So who gets married to Isabella, the heiress / Queen since her half-sister Sibylla died at the siege of Acre? Possibly her competent next husband after Henry in real life , Amalric of Lusignan - the younger brother of Guy of L, Sibylla's incompetent husband and the man who lost the Kingdom to Saladin in 1187. Guy is hated too much to get his throne back - and if there is no Richard to march South and take Jaffa and improve the size of the Kingdom , the Crusade will end once Acre has fallen as Philip was keen to get home.
A smaller Kingdom will exist, stopping short at Acre and Mount Carmel, and will be easier prey for the Ayyubids in later decades - and not last until the Mamelukes take over as in real life, unless the remaining lords in the coastal strip N to Beirut can get European help to expand. (Possible if the Fourth Crusade does not go to Constantinople - ie if Theobald of Champagne does not die early so he is in command as originally planned.) Unless Richard's army can take Cyprus in revenge after R dies or he dies of a wound after the fall of Nicosia, Isaac Comnenus keeps Cyprus - but he has no son and the Greek mini-state is in rebellion against the Byz Empire so either Isaac II reconquers it c. 1194-5 (instead of continuing his Bulgarian war so he does not mess up the latter and get deposed by his brother?) or it survives until either the Venetians or refugee Crusader lords from Acre grab it. I can see the Crusaders thinking Cyprus a useful refuge, full of rich lands to rule, and safe from Moslem attack so taking it over c. 1200 to 1210.
Finally, if John is king in 1192 not 1199 does he still divorce his wife Isabella of Gloucester, who was his cousin so the Church was trying to invalidate the marriage as illegal and allegedly J did not have sexual relations with her so declaring it void was easier - they had no children. If John remarried at this date , his OTL second wife Isabella of Angouleme (b either c. 1185 or c. 1188) is too young so they do not marry - and her ex-fiance's family, the Lusignans led by Amalric's nephew, do not defect to Arthur and Philip in revenge and make their task of seizing Poitou/ N Aquitaine easier. If John remarries and has a son, this person may be adult or at least a teenager by the time that the English barons have enough of John and his taxes and murders - so do they try to put him on the throne in the 1215-16 revolt , not ask Philip's son to lead them?