John of Gaunt gave the claim to Castile because a year after he left a group of English lords (called the Lords Appellant) seized power from the royal court. John returned to England and allied with the King, spending the rest of his life rebuilding royal power, leading to the destruction of the Lords Appellant in 1397.
Something needs to be done to maintain royal power in England, so that John of Gaunt can pursue his claims in Castile without distraction by politics in England. Basically, I think he needs England to remain stable so that his possessions in England can continue to finance is claim on the Castilian throne. Chaos in England means an end to those funds. If this is accomplished, then John of Gaunt can engage in a nice long, costly war with the King of Castile.
If John of Gaunt is engaged in this enterprise, then I think that his son, Henry Bolingbroke (Henry IV of England OTL) would probably come down to Spain to fight with him. In OTL Henry went to war in Lithuania in support of the Teutonic Knights, and went to Jerusalem on a pilgrimage. So in this TL his foreign trips are made to support his father.
The really big effect that this will have is if John of Gaunt wins the Castilian throne. Henry would be his logical heir, which would really throw the politics of England into a whole different set of circumstances. Without Henry will someone else act to overthrow Richard II, and thus begin that most popular and confusing of English wars, the War of the Roses? Who will lead the Lancasters if their leadership is the Royal family of Castile? How will everyone's favorite legitimized bastards the Beauforts come out of this royal rumble?