If either side ever wanted a war, there were plenty of things that could have led to war: Fashoda for one, the Siam crisis was even more intense. But the list is long Niger, Madagascar, Tunis
They don't fight because wars don't erupt over incidents but because of deep political conflict. One side wants war and then the incident is used. America wanted war and the Maine explosion was the flimsiest of pretexts. Meanwhile, something like Dogger Bank is just a blip on the screen because neither Russia or Britain saw war in their interests
Getting to a situation where either side was determined for war is rather hard. Post 1870, France would need a powerful ally to keep Germany at least neutral which is hard. From 1848, Napoleon III whole policy is based on good relations with the British- to the point that he surrenders at Sedan rather than sneak out through Belgium
That leaves the 1816-48 period. Frankly, by 1816 Europe was so sick of war I doubt if anyone would start a new one for quite a while but France is still immune from an invasion so its possible but extremely
To get an Anglo-French war you would have to radically change the course of Europe. A French victory in the 1870 War leading to insane demands. But that would also mean no Napoleon III