I’d have to argue against both Trafalgar and the Battle of France.
In the case of Trafalgar, while a French/Spanish victory would have been a major immediate upset for the British, I think the general superiority of the Royal Navy was up to a quick recovery, (and the loss of ships of the line would not have been as disasterous as the loss of an equal number of frigates). The Channel Fleet was still in existence, so I think it would have been similar to the naval defeats the British suffered in the early stages of WW I, none of which, including Jutland, affected the main British fleet’s control over the western European approaches. Neither the French nor the Spanish could afford to make a major increase in the resources devoted to their fleets, while Britain could, and would. More to the point, however, I believe that it would have been very difficult for the British to lose the battle given their general superiority in training and abilities, and the best outcome for their opponents would have been to not lose as badly.
For the Battle of France, a German loss would certainly have had a major effect on WW II, but, again, I just don’t see it happening. The French were likely incapable of pulling off a victory once the Germans poured through the Ardennes, given the underlying factors which lead to the quick collapse of the French Army, and the British had nowhere near enough forces to seriously affect the outcome by themselves. I don’t think that a battle in which it is so unlikely that the outcome would change is a serious candidate.
I would agree with Hastings, as it was very much a near-run thing, and a loss by William would probably have resulted in the majority of his forces buggering off back to France. Don’t forget that his army was not a single entity, but a collection of his own feudal troops and those of other feudal lords and younger sons who joined up in the hopes of loot and land. If he lost, even if he survived, would the invaders have held together for a second try or just done some local looting and rembarked. And if he lost once, would he have been able to assemble an army for a second try?