Okay, then, Prince Henry, Duke of Gloucester. His eldest will probably NOT die in a plane competition, because as heir, he won't be allowed to compete because of the danger. The Duke of Kent will probably be recalled from active duty and survive to sire more children with Marina. Gloucester and his wife might make an effort to have more children. Hitler will make more of an effort to obtain custody of the Duke and Duchess of Windsor; if he succeeds, Parliament will specifically pass legislation barring them from the throne (there were already rumors about the Duke, the Duchess, and their pro-Germans views), on top his renunciation of it in 1936.
The Duke of Windsor is already out of the line of succession. Part of the deal surrounding the abdication was that he was required to renounce his rights to the crown and all royal titles not just for himself but for any heirs too - in effect, he was stripped all the way down to a commoner before being given the title of Duke of Windsor by George VII as a new creation. There is no need for new legislation as it was all dealt with at the time -
His Majesty's Declaration of Abdication Act, 1936. It's one of the peculiarities of the British constitution that the monarch has enormous powers in theory, but very few in practice - he couldn't even give up the job without parliament's approval (the king abdicated on 10 December but it was of null effect until he gave Royal Assent - as king - to the bill a day later).
As to what happens, we get King Henry IX, the British get slightly more pissed off and determined to finish things, but otherwise the course of the war probably doesn't go much different.
The really interesting thing will be the immediate aftermath. The death of the king, queen and both of their daughters would normally require a spectacular funeral complete with a lengthy lying in wait and Westminster Abbey full of dignitaries. This is obviously going to be problematic in wartime! If the Germans have any sense they will make a big deal of declaring a 72 hour moratorium on air raids in the London area to allow the funeral to proceed in peace - I would think this would certainly be Goebbel's advice (he would be more aware than most of how big a propaganda disaster this could be), but Hitler might give in to the temptation that so many high value targets in such a small space would represent.
The question then becomes what the British do - stage a scaled-down funeral quickly and run the risk of appearing scared, stage a full scale funeral somewhere safer (Edinburgh would be an interesting choice propaganda wise) or just go "Screw you" and set up a full blown service in Westminster Abbey and move virtually the whole of Fighter Command to the south of London. Churchill would be tempted to go for the latter, but I suspect wiser heads would prevail and we'd get some form of the second.