Greetings! This is an interesting question you have put forward, and it actually makes a fairly good point (at least on the surface) about the historical ties (or lack thereof) between Great Britain and France. There are however, also deeper reasons and causes as to why the British ended up signing the Entente Cordiale with France in 1904 (which by the way, was NOT the military alliance that some tertiary sources claim it to be, that would come in later revisions to the Dual Entente Alliance). Let's begin.
The German Question
Before the 20th century had even dawned on Europe, the Palace of Westminster and the House of Hohenzollern had cordial and even warm relations between them. Germany's Kaiser Wilhelm II often made trips to Great Britain, where he reviewed the Royal Navy and British Army, whilst also calling for a certain "Kultur" (culture; close tie) to develop between these two great powers. The British meanwhile, viewed Germany as a potential ally (though by no means took steps to ensure this) should Russia attempt to expand into the Balkans once more. Remember, at this stage of the late 1800s the main British concern was with their Empire, and particularly the security of both the Suez Canal and the British Raj. In the case of the former, the "Sublime Porte" (the Ottoman Empire) served as a useful (if at times troublesome and unstable) "obstacle" which the Russians had tried (and failed, though at heavy Anglo-French-Ottoman cost) to surmount in the Crimean War of 1854-56. The latter however, the "jewel in the crown of the empire", was also threatened by Russian expansion into Central Asia (Persia in particular, with an eye for the overland trade routes into the Indian subcontinent). As Secretary of State for Foreign Affairs and Prime Minister Lord Palmerston put it in 1835:
"Sooner or later, the Cossack and the Sepoy [Indian soldier serving under Britain], the man from the Baltic and he from the British islands will meet in the centre of Asia. It should be our business to make sure that the meeting is as far off from our Indian possessions as may be convenient."^(1)
Of course, when Prussia became Germany and quickly became a significant power in Central Europe, Russia remained a threat, constantly eyeing the Balkans (which by the late 1800s, had by some extent gained independence from their former status as Ottoman provinces). This was the atmosphere in which Germany appeared to be a reasonable option to ally with, as the French were still viewed suspiciously after the previous "Great War" (yes, the Napoleonic Wars were called that by the British until the First world War). However, the British were also aware that they were not particularly well viewed in Berlin either. The "Iron Chancellor" of Germany Otto von Bismarck often used William Gladstone's Britain as "an example of the democratic depths to which Germany might sink if they relaxed their vigilance". The German industry and economy was also a rising threat to British commercial interests, though I highly recommend going into the specifics of this further yourself if interested.
Then came a new Kaiser, Wilhelm II, and in 1890 Bismarck himself was no longer a key diplomat that either side could rely upon (having been dismissed that year by the far more aggressive-minded Wilhelm II).
1897 - Turning Point
In the critical year of 1897, the Germans began to depart from their previous position as the orchestrator of the "Concert of Europe", and were far more imperialist in their foreign policies. Under Wilhelm II, the Germans began building their own empire abroad in Africa, the Pacific, and strengthening their economic power at home (by 1914, the furnaces and mines of Germany had already outproduced the British in steel and coal, an impressive feat). This was all done under the so-called mantra of, Weltpolitik *(*world politics), in which the militaristic Prussian state led Germany in becoming a Weltmacht, (world power). This was all brought to the forefront of European politicians with the following line delivered by Secretary of State of Foreign Affairs Bernhard von Bulow in December of 1897 to the German Reichstag (essentially their parliament).
"Mit einem Worte: wir wollen niemand in den Schatten stellen, aber wir verlangen auch unseren Platz an der Sonne." (in one word: We wish to throw no one into the shade, but we also demand our own place in the sun.).^(2)
In this light, the British began to view the Germans with increasing suspicion. It did not help that shortly after such aggressive expansion and imperialist policies, Wilhelm II started a naval arms race with the British, something which puzzled and troubled Whitehall. After all, why did the Germans - who already possessed (at least from the British point of view) the most powerful army in the world - require an ocean-going navy?
To be fair, the French were not particularly open to Britain either at this time. They were still concerned with the occupation of Egypt since the British had moved troops into the former Ottoman province in 1882. By 1904 however, that issue had mostly been settled by the French accepting that Britain would remain in Egypt for quite some time, though this new "neighbour in North Africa" would not threaten their own colonial holdings in French Algiers and Morocco.
In 1892, the French and Russians concluded their own alliance, the Franco-Russian Alliance (I discuss this in more depth, as well as why the Germans did not ally with the Russians either, in this thread) This left Britain practically isolated in terms of allies, as the Germans had already united with Austria-Hungary and Italy in the Triple Alliance. So in 1904, the British government decided that putting aside their historical rivalry and antagonism of France was the better option, and thus was born l'entente cordiale, with Russia joining in 1907 to form the Triple Entente, after her humiliation in the Russo-Japanese War of 1905 convinced both St. Petersburg and London that an alliance would be best to counter German-Habsburg ambitions.
Conclusion
So to answer OP's question: the Anglo-German Alliance, though once contemplated seriously as an option for both powers, was never formally signed or negotiated due to concerns that Wilhelmine Germany, under the rhetoric of Weltpolitik, would collide with Britain and threaten the security of her empire to a larger extent (at least in the 1890s and 1900s) than either France or Russia could. Hope this helped with your question, and feel free too ask any follow-ups on related matters of international relations before World War 1.
An overtaking of the French threat with the German certainly is one of the main reasons behind Anglo-French rapprochement in the late 19th century which /u/Starwarsnerd222 has covered really well, but there were other dynamics at work too.
Colonial Origins
The British Empire in the late 19th century felt under pressure from just about everywhere — there were economic concerns at home, continued debate over home rule for Ireland, perceived French threats to Egypt and the Far East, and Russian pressure on the Northwest Frontier in Afghanistan to name just a few of the governments worries — and there simply was not either the resources or political will to deal with all of them. The British very much wanted to avoid becoming involved in a major war with another European power which could hardly afford, but also needed to ensure the protection of its colonies. Throughout the 1890s this led to mixed approaches to foriegn policy, in some cases negotiation— such as over Siam in 1896 — and in others a brinkmanship approach — such as during the Fashoda Crisis of 1898. Both policies seemed viable and there was substantial domestic debate over the best course of action.
However, following the humiliating and ruinously expensive Second Boer War and with the perceived need to concentrate more on home defence with potential threats from both France and Germany nearby the isolationist urge to enforce British policy rather than negotiate it was increasingly discredited. Accordingly, a diplomatic agreement that could offer long term security for Britain’s colonies became increasingly attractive. Infact, the actual text of the Entente Cordiale, which would be the basis for later cooperation, is largely spent adjudicating colonial disputes.
Meanwhile on the other side of the Channel, the main problem for French foreign policy from the formation of Germany in 1871 right the way till 1940 was that of security and as a rule she was generally eager to seek out allies to help ensure it. In the case of the late 19th century the French foreign minister at the time very much favored building a relationship with Great Britain as a safeguard against German aggression and pushed heavily for peaceful solutions to colonial troubles and the general easing of tensions described above.
So, a situation developed where Britain needed to reduce the chance of conflict along her Imperial borders and France was willing to make concessions to do so in return for improved relations.
Russia and the Rise of Military Partnership
In understanding how a limited treaty mostly concerning colonial matters into a guarantee of security we need to look at Russia. The 1904-5 Russo-Japanese War profoundly weakened the perception of Russia's capability and thereby also weakened her role in the Franco-Russian alliance as a counterweight to Germany
For a decade, until 1904-5, the Franco-Russian alliance, alongside the German-led Triple Alliance, had been the key variable in the calculations of the chancelleries of Europe. The alliance acted as a kind of vice on Germany, restricting the latter's diplomatic freedom of manoeuvre in Europe...In 1905 the Franco-Russian Dual Alliance was effectively disabled: the vice had cracked open.
The same year Germany sparked the First Moroccan Crisis as part of the Weltpoltik discussed elsewhere. Without the assurance of Russian support the British worried the French would seek a settlement with the Germans which would greatly endanger British interests, so Britain felt compelled to support France. Lord Grey, the foreign minister, made hinted to the French that, while he could not promise anything more than neutrality in the event of a war public opinion might force Britain’s hand and authorized informal planning talks with the French military attache. Though this didn’t result in any firm commitments it did represent a fundamental shift in British policy going forward. The new common thinking in British foreign policy can be summarized fairly easily:
If, however, the entente were allowed to collapse, Britain would be internationally isolated, and thus vulnerable to pressure by other powers. Sir Charles Hardinge, the permanent undersecretary, endorsed Grey's argument: 'If France is left in the lurch an agreement or an alliance between France, Germany and Russia in the near future is certain. This ... is the Kaiser's ideal, France and Russia becoming satellites within the German system.' If,however, the entente remained intact, then Britain's position as the linchpin of European politics would also be secure
DISCLAIMER: I am not saying this is the only reason Britain entered WWI, that is a much bigger topic than can be discussed here
Conclusion
While concern over the rise of Germany certainly helped drive Britain and France together the need to settle colonial questions also played a large role and was, perhaps, the only purpose of the improved relationship in 1904. However, the removal of Russia as a force able to check German ambitions drove Britain to step into the gap and tie herself to French Security.
Follow Up Answer - Monarchy, Democracy and Republics
I can also shed some light on your follow up question on why a conservative monarchy and a revolutionary republic might work together. In the period being discussed the United Kingdom was a constitutional monarchy and while the sovereign had more power, both official and simply by way of influence, than today their role was much closer to our modern understanding of the royalty than the power their ancestors wielded in earlier centuries. Britain saw itself as a democracy and saw the German Empire very much as a militaristic and totalitarian state. Indeed, Prussianism virtually became a synonym for such things. By contrast, the French Republic was a liberal democracy similar to the UK. For more detail on both I'd point you to /u/Starwarsnerd222 and /u/endgame00 's answers.
Works Cited
Brailey, Nigel J., ‘Anglo-French Rivalry over Siam and the Treaties of April 1904’, in Anglo-French Relations, 1898-1998: From Fashoda to Jospin, ed. by Philippe Chassaigne and Michael Dockrill (Palgrave Macmillan, 2014), pp. 56–68
Darwin, John, ‘Afterword: The Imprint of the War’, in The Impact of the South African War, ed. by David E Omissi and Andrew Stuart Thompson (Houndmills, Basingstoke, Hampshire; New York: Palgrave, 2002), pp. 289–302
Kennedy, Paul M., The Realities Behind Diplomacy: Background Influences on British External Policy, 1865 - 1980 (London: Fontana Press, 1985)
Otte, T. G., ‘“Almost a Law of Nature”? Sir Edward Grey, the Foreign Office, and the Balance of Power in Europe, 1905-12’, Diplomacy & Statecraft, 14.2 (2003), 77–118
———, ‘From “War-in-Sight” to Nearly War: Anglo–French Relations in the Age of High Imperialism, 1875–1898’, Diplomacy & Statecraft, 17.4 (2006), 693–714
Venier, Pascal R., ‘Théophile Delcassé and the Question of Intervention in the Anglo-Boer War, October 1899–March 1900’, in Anglo-French Relations, 1898-1998: From Fashoda to Jospin, ed. by Philippe Chassaigne and Michael Dockrill (Palgrave Macmillan, 2014), pp. 44–55