Hopefully this won't be deleted, but I'll try. I've never read anything specifically discussing relations between France and Britain at that point in time, so everything is couched in the context of other contemporary events I know about. In The Crimean War: A history, Figues does not go much into the basics of the relationship, but overall the picture he paints is of a resurgent France under Napoleon III indirectly pitted against and shadowed by a triumphant, wealthier, more democratic and more powerful British Empire. By then Waterloo was still remembered obviously but you don't get the feeling that Britain was all that stuck in the original Napoleon drama - rather, you see two powers bidding for increased influence in the continent and beyond, forming an alliance of convenience against a perceived greater threat (Russian expansionism) that is shaped very much by special interests manipulating public opinion, more so in Britain than in France. There was also an important latter-day religious component to this, and despite the fact that France was largely Catholic and not necessarily friendly to the Church of England, they were both ultimately Christian and putatively concerned about the treatment of Christians in the Ottoman Empire and opposed to the spread of the Orthodox faith, while at the same time pushing for more open markets and trade (by then Britain was completely sold into the Free Trade thing, in a parallel to the US nowadays) which would have been hindered by Russian control of the region (see for example Millman's Britain and the Eastern Question). All this while attempting to prop up the Ottomans to prevent them from collapsing and throwing the region into chaos.
In general, I would say France is largely presented as a slightly has-been empire, jumping at the chance to go along with the British on an adventure that would give Napoleon the military glory he desired without having to march on Prussia or some such dangerous folly. On the other hand, most military assessments I've read give the French a much better grade than the British, who were notoriously lax in the logistics and equipment departments.
There is no difference there in my view than former Warsaw Pact nations joining NATO or German troops deploying to Afghanistan to help the Coalition effort. Immediate convenience and national interests are more important than century-old grudges.
To piggy back on the rest of the comments, English European policy was never really focused on one country so much as preventing any nation from getting too powerful on the continent. For a long time France was the final boss on the continent, but thanks in part to the rapid development of its agriculture and the demographic devastation of 22 years of almost non stop conflict, it's population started plateauing much earlier than most. Between 1800 and 1900 it only grew from 26 million to 40 million, meanwhile the rest of Europe, especially England and Russia were booming. This results in Russia becoming the new perceived most powerful nation on the continent, with by far the largest army. For example, in 1848 it was Russian troops who saved the Austrian empire from internal rebellion. Along with its push into central Asia threatening the crown jewel of the United Kingdom's empire, India, this was what pushed England and France together.
Sources: Figes, The Crimean War: A History and Keegan The First World War