I will make the argument that England and France were both mostly looking out for their own interests in the period of 1871-1914, and that their respective interests compelled the French Republic and the UK to form the Entente Cordiale in 1904.

First, it is important to note the obvious -- that there was a new, major European power (Germany) on the scene in 1914, which had not existed in 1815. Prior to the creation of a unified German state in 1871, Prussia and the Austro-Hungarian Empire were the two dominant forces in Central Europe, with the Russian Empire, France, and the British Empire playing similarly large roles in continental politics and diplomacy. Prussia and France never got along, for a whole host of reasons, most of them dating back to period of the Napoleonic Wars. When the unified German Empire formed, it essentially inherited Prussia's diplomatic stances, as the Hohenzollern King of Prussia was chosen to be with the implicit threat of force Kaiser. The German Empire was founded in a series of very fast, expansionist wars, where Prussia led a German coalition against Denmark (1864), Austria (1866) and France (1870-71), afterwards referred to as the Wars of Unification. The most important thing to take away from these wars, with respect to your question, is that France was purposefully humiliated. The German Empire was declared in the Palace of Versailles, where the Kaiser was crowned by German princes. This act was taken both to legitimize the unification as a fait accompli (if the empire was born out of a victorious war over the continent's previously-dominant power, in the palace most symbolic of monarchy, who would stand up to contest it?) and as a deliberate move to antagonize the French. Additionally, the German Empire annexed Alsace-Lorraine, an economically-significant region, which France never gave up on demanding back, and which was taken with relative impunity. Any hopes of the nascent German Empire making nice with France were hopeless from this point on. After the wars of unification, Bismarck made a very sincere (as far as we can tell) attempt to assuage the leading European nations that Germany was no longer going to expand, now that it had been formed. Meanwhile, France adopted a foreign policy of fierce revanchism (essentially, telling any other country that would listen about how they had been wrongly bereaved of Alsace-Lorraine in the Franco-Prussian War, and how they deserve some vengeful justice in the matter).

From the French revanchist perspective, war with Germany was always inevitable, it was simply a matter of when. However, the French could not alone overcome the German military. It needed an ally to support any revanchist hawkishness. Initially (talking about 1871), the British Empire was not much involved in the politics of the continent, in line with its traditional policy of "Splendid Isolation", whereby British interest was more invested in international expansion than European wars. While the formation of a German state greatly changed the balance of powers, which had been established at the Congress of Vienna in post-Napoleonic-Wars Europe, the British were wary to get involved with any side. France's other candidate states for military alliances were the Russian Empire and the Austro-Hungarian Empire. This was where Bismarck's foreign diplomacy skills, and experience as a former ambassador to Russia and France for the Prussian state, came in handy with thwarting French overtures. Bismarck adopted a policy of isolating France, to the fullest extent possible, from Russian and Austrian courts. Bismarck made favorable deals with the Austrians (who, remember were at war with what would become Germany as recently as 1866) and the Russians, sometimes putting Germany at short-term disadvantage, just to maintain working relationships that could be leveraged to keep French revanchists out of the discussion. This was a circus act of diplomacy that Bismarck had juggled masterfully, and that would be masterfully undone in the decades after his forced resignation in 1890. This is because Bismarck, however brilliant a diplomat, never convinced either his succeeding Chancellors or, more significantly, the brash Kaiser Wilhelm II of this policy's utility. Wilhelm II, a man of questionable leadership characteristics (it's still debatable whether he was genuinely crazy, or just impulsive to an effectively insane degree), committed a series of diplomatic blunders that rapidly unravelled Bismarck's systems of shifting alliances. While Bismarck had nightmares about a two-front war with Russia and France, Wilhelm believed that his and the Russian Tsar's common monarchical existence would be enough to secure their alignment against the Republicans of France -- a decidedly Napoleonic-Wars-Era way of thinking that would not hold water. Accordingly, Wilhelm rather punitively invalidated previous diplomatic arrangements with Russia (who promptly became friends with France) at the first opportunity, in order to favor Austro-Hungarian interests over Russian intersts in the Balkans.