Well, the change was not night-and-day. The French stopped being hated enemies after the Napoleonic wars and over the course of the Pax Britannica. After 1815, the British became the biggest power in Europe, and European policy in general started focusing away from the continent and more overseas. It was the age of empires, and a period known to many as the Pax Britannica.

The English established their 'Second Empire', mostly concentrated around Canada, Australia, India, and laterally eastern Africa. The French concentrated on Indochina and northwestern Africa. Both sides were expansionist, but found that cooperation was more helpful than confrontation. For example, when Napoleon III became Emperor of France in 1848, the British were worried because he had the intention of expanding the French Empire. Lord Palmerston was the foreign minister at the time, and he had close relationships with many leading Frenchmen, including Napoleon III. His goal was to make sure that relations were friendly in order to focus Britain's efforts on other more important parts of the world. Napoleon III, by his turn, was quite pro-British, partly because he did not want to upset the Imperial powerhouse, but partly because an allegiance between France and Britain was a very powerful force of cooperation.

The benefits of this allegiance showed themselves during the Crimean War and the Second Opium War. In the Crimean War, the Brits and the French chose to fight the Russians in order to check their power. Napoleon III tried to organize an alliance with Russia in order to dominate Europe, but the Brits found out and it was abandoned. In the Second Opium War, the French and British co-operated in order to force the Chinese to sign a beneficial treaty.

The late 19th century was a little less friendly. Tensions in Africa very nearly led to war, but at the same time Germany was becoming more and more powerful, so British and French policy aligned once more. This is the period if the Entente Cordiale, and eventually led to the First World War.

It's worth noting that, even though this period saw France and Britain as allies, they still weren't on super-friendly terms. It was mostly a political allegiance, based on the fact that they had similar goals in the world. They understood that they were both the biggest imperial powers in the 19th century and that stepping on each other's toes would be less productive than if they banded together. In Britain, France was a key part of the 'balance of power' in Europe. And in France, Britain was too powerful to be enemies with, so the most productive strategy was co-operation.