Interesting. So a war of succession would be essentially inevitable?
It really depends on the regency (and later reign, if we get that far) of Richard II. A very stable regency could happen, dominated by the late king's senior advisors with the support of the Church and the Lords. OTL, the Archbishop of Canturbury (Simon of Sudbury) was also Lord Chancellor, placing him in a prime position to dominate the regency in the absence of uncles. The problems leading to Wat Tyler's Rebellion (which killed Simon IOTL) were a combination of attempts by Parliament to guard against a coup by John of Gaunt and by unpopular taxes needed to pay Edward III's war debt -- the former would be mooted ITTL, but the latter would still be necessary, so the rebellion may or may not be mooted.
Without the rebellion, Richard is likely to defer at least somewhat to his advisers and Parliament, since the rebellion killed off some of the stronger leaders and also seems to have served as a bit of a turning point in Richard's views of his own role as King. He might even make it through his entire reign without being overthrown. The big question is whether he manages to produce an heir of his own, or if he eventually dies without producing an heir, producing a succession crisis, but at least leaving a strong government to manage the transition (probably either crowning whoever is the leader of government at the time, or hand-picking a suitable puppet king who has a few drops of Plantagenet blood).
On the other hand, if Wat Tyler's rebellion does still occur, Richard's absolutist inclinations are likely to be even nastier than IOTL, since there wouldn't be any clear rival claimant for opposition to coalesce around. If Richard does die without heir, or if he manages to provoke a successful rebellion despite a lack of semi-legitimate rival claimants, expect an unholy free-for-all, as Richard's overthrow would set a precedent for an "occupative" principle of succession (i.e. the throne passes to whoever is strong enough to seize it).