The death of a prince of the blood would take a considerable amount of shine off of Richard's quashing of the rebellion. This has immediate consequences for the government of the realm. Firstly, Richard assumed personal control of government in the wake of the revolt, despite being just 14. This was acceptable in part because the "permanent council" system that had been set up after Edward III's death had been a miserable failure and also in part because Richard came away from the revolt looking like a hero to the nobility. That's not the case here and might mean that the council system continues on into the mid 80s.
Secondly, and most dangerously for Richard, Gaunt may come away from this blaming Richard for Henry's death. Gaunt protected Richard throughout his life. Richard's childhood reign 1381-1386 is basically a series of incidents in which Richard fucks up badly, everyone wants to get rid of the kid and/or the people around him, and then Gaunt and Joan of Kent swoop in to calm everyone down. It's not a coincidence that there's a rebellion almost the moment Gaunt and Joan are out of the picture or that Richard, after two years of political impotence, is able to overthrow the government of the Lords Appellant upon Gaunt's return. But does Gaunt bother to play this role in ATL? Or does Gaunt let Richard destroy himself even more quickly in ATL?
He might attempt to legitimise the Beauforts or try to have another son with Constance of Castile to make up for that (as of now the latter is more likely).
Gaunt repudiated Katherine and reconciled with Constance after the revolt in OTL. They may have tried having more children after that, as there is some evidence (though it is very slim) that Constance was pregnant in 1386. I think it is a guarantee that the two try for more children in ATL.
Mary de Bohun would possibly go to the Convent here, as Thomas of Woodstock wanted (which increases his power as he gets the full Bohun inheritance).
I suspect that Richard may try to push the girl onto his landless half-brother, John.
How would this affect the succession? Would Richard declare the Mortimers his heirs earlier? What else might change.
The Mortimers get an awful lot of attention in retrospect thanks to the Wars of the Roses, but there's almost no actual evidence that Richard wanted them as his heirs. On more than one occasion Richard tried to destroy the Mortimers, just as he did the Lancastrians.
edit: I'll clarify, there is a chronicle reference to Richard announcing the fourth earl of March as his heir, but there's no government record that he actually changed the line of succession in this way -- no act of parliament, no letter patent, nothing. Considering that the parliament in question is 1386, it's clear that the pronouncement before parliament was only done to counter the threat to depose him (i.e., to threaten the ascension of an even younger boy king).