Henry IV was the son of John of Gaunt, and Blanche of Lancaster, not Henry III, neither of them were the King and Queen of England, Henry IV usurped or displaced his cousin, Richard II, this was the starting point of all the troubles for England in the latter part of the 1400's now known as the war of the roses, because in a strict order of succession a younger cousin Edmund Mortimer should have been next in line for the crown. Later descendants of Edmund would base their claims to the English throne through him.
Another King, Edward IV was rumoured to be a bastard, this was mostly put out by his enemies, and those who wished to discredit the eligibility of Edward's children to inherit the throne, but some people still believe this. His mother Cecily Neville in her will explicitly stated that Edward IV was the son of her husband and herself. While I don't believe that Edward IV was illegitimate, if this was so, then in order of succession the rightful claimant to the English throne is the eldest living child of Mike Hastings who died in 2012, he emigrated to Australia and was a fork lift driver, there was a documentary about this on TV a few years ago.