1915. A year is all it takes.
First of all, I should say that I'm dealing with two imaginable scenarios for why WW1 doesn't break out from the July Crisis.
(A) Gavrilo Princip gets food poisoning on the way to Sarajevo. Franz Ferdinand finds that the 28th June, 1914, is nothing more than the day of a rather boring state visit. The July Crisis never happens at all.
(B) Franz Ferdinand still gets shot, but at a certain fateful time in the summer of 1914, Emperor Wilhelm II is
not on a cruise (as he was IOTL) but, rather, in Berlin. So when Wilhelm II reads Serbia's response to Austria-Hungary's ultimatum, decides that it's enough for him to count it as a victory and war is therefore unnecessary, and writes a letter to Emperor Franz Josef of Austria to express this opinion, Wilhelm doesn't send the note via his foreign minister Jagow who then decides not to send it (which is what happened IOTL), Wilhelm sends the note straight to Franz Josef. Austria-Hungary doesn't dare to go to war against Serbia without German support.
The situations produce rather different outcomes.
_________________
In Case A, the United Kingdom and Russia drift apart over the issue of Persia and the Anglo-Russian Convention is not renewed when it comes up for renewal or expiry in 1915. This does
not mean that the UK is now Germany's best friend, only that it's Russia's rival. The UK sells its ships to the Ottoman Empire, and, after the expiry of the Anglo-Russian Convention, starts to prop up the Ottoman Empire against the Russians.
France is now in the difficult position of being very friendly with the UK and very friendly with Russia, at a time when the UK and Russia are very definitely not at all friendly with each other.
Serbia is still sending in nationalist agents to stir up dissent against the Austro-Hungarian authorities in Bosnia, and Bulgaria and the Ottoman Empire are still really annoyed at the outcome of the Balkan Wars. So a Balkan catalyst leading to war, in this situation, is still quite probable.
We now have two possible outcomes. If the UK isn't dragged in straight away, the UK will have no interest in helping Russia, but if Germany attacks Belgium or if Germany looks like it might use its naval might against France, the UK will still act against Germany to prevent that. That gives an OTL-esque outcome in terms of who is allied with whom. But it's also possible that the war starts off in South Asia or East Asia, with the Anglo-Russian clashes there. In that case, the UK will be opposed to Russia from the start. So we could see a war of UK + Ottoman Empire + Japan vs Russia, with France, Germany and Austria-Hungary neutral.
_________________
In Case B, Austria-Hungary is outraged, though not entirely surprised (in this era IOTL they were worried about how lukewarm German support for them was), by Germany's failure to support it in what it considers a matter of the highest possible importance. (A Serb killed the heir to the throne when the Emperor is the main thing holding Austria-Hungary in the first place and when the Emperor is very old and will die soon; it's difficult, from that perspective, to see it as anything other than a threat to Austria-Hungary's very existence.) Austria-Hungary is exceptionally unhappy with Germany, but still unfriendly with Russia too, for obvious reasons.
The Anglo-Russian Convention expires, as in Case A, and the UK props up the Ottomans, as in Case A. There is now an Anglo-Ottoman bloc opposing Russia and its Serbian client state. Austria-Hungary is likely to join this bloc; the United Kingdom is Austria-Hungary's natural ally if both of them see Russia as their great enemy. I don't think it takes much of a stretch to see the UK propping up Austria-Hungary as a bulwark against Russian expansion (Russian expansion, especially in Europe but also in Asia, was viewed really badly by the British). In the case presented here, there's an extra benefit: the UK was really not fond of the Serbian regicides, and, without the breakout of WW1 changing things, is likely to be sympathetic to Austria-Hungary and antagonistic towards Serbia.
Germany has just monumentally screwed up and lost its only major ally, and France and Austria-Hungary are obviously not options. So who can Germany turn to? It can turn to the United Kingdom, but that'll probably fail for the same reasons why Anglo-German alliance failed IOTL. Alternatively, it can turn to Russia. The fall of Germany's alliance with Austria-Hungary ends the main reason for the Franco-Russian Alliance; Russia has little interest in opposing Germany for its own sake.
Russia may stay loyal to the Franco-Russian Alliance anyway; it certainly had plenty of supporters in important places in Russia. In that case, Germany is isolated but is unlikely to go to war against anyone; colonial issues were largely settled by this time and France, though wanting Alsace-Lorraine back, isn't just going to invade without a
casus belli. A Balkan, South Asian or East Asian war between the Anglo-Austro-Ottoman bloc and Russia is quite possible, but in that case France has no real reason to get involved on either side; France will probably bow out. Russia would almost certainly lose such a war. As for what happens beyond that, someone can write a TL about it if they want to; I'm not going to go that far into the future of this brief idea (unless asked to).
Russia may, alternatively, ditch France in favour of Germany (there were supporters of this course of action in important places in Russia IOTL as well). In that case, France finds itself on the side of the UK, the Ottomans, Austria-Hungary and Japan, opposed to a Germano-Russian bloc. Italy wants territory from both France and Austria-Hungary, but an Anglo-Austro-French alliance leaves the Mediterranean so unambiguous that Italy would never attack them until/unless the war is already pretty much decided in favour of the Germans and Russians.
The Anglo-German naval race ended in 1912:
https://en.wikipedia.org/wiki/Anglo-German_naval_arms_race#The_race
Basically the British navy used the 'race' to get more funding after the advent of the Dreadnought made all other BBs useless. For the Germans it was more about establishing superiority over the Russian and French combined fleet, while ensuring they couldn't be close blockaded by anyone. By 1912 the Army League usurped the Navy League's position in German politics, so the navy ran out of support for any more naval expansion. The British government had an anti-German clique in charge after the events of the early 20th century, but as the German threat recedes in terms of a naval challenge and the Russians become more threatening, then the British government will become less anti-German, though rivalry in terms of economics would remain.
Admittedly not having read the Wikipedia article, I presume the gross simplification is Wikipedia's fault and not yours.
Describing the British government in 1914 as an "anti-German clique" is very much a stretch. Sir Edward Grey could definitely be described as anti-German, but not so much the rest of the government, including the Prime Minister. It was a very particular set of events that allowed Sir Edward's view to prevail over the views of many others in the Cabinet.
Nor did the new advances in naval technology, with dreadnoughts and battlecruisers, make all existing battleships useless (though I've seen this idea many times elsewhere). Indeed, I'd pick a pre-dreadnought battleship over a battlecruiser any day; the battlecruisers performed abysmally, at least how they were used. And dreadnoughts, though superior to pre-dreadnought battleships, weren't so superior to pre-dreadnought battleships that just a few of the former could destroy literally dozens of the latter—which is what would be necessary to make the UK's existing naval advantage "useless", rather than just "somewhat reduced". That especially counts for the dreadnoughts which were around early in the dreadnought era; obviously the later dreadnoughts were superior to pre-dreadnought battleships by a larger margin than the earlier dreadnoughts were.
The Royal Navy used the race to get more funding, but that had nothing to do with the dreadnought or with Germany; it had everything to do with the Royal Navy's habits as they have lasted for a very long time, i.e. pick an enemy country and inflate their capabilities so that they sound like a great threat, in order to get more funding. Just look a little earlier than the naval scare about the German navy in the British press and you will see a remarkably similar naval scare about the French navy in the British press. (This, by the way, is why it's so foolish to draw the conclusion that the Anglo-German naval arms race made war between the UK and Germany inevitable; no, it didn't, it was just typical behaviour, and we only tend to ascribe more significance to it than we do to other instances of the same behaviour because we know that IOTL the UK and Germany did go to war.)
The Germans conceding the arms race is true to an extent, but only somewhat; Germany's naval construction was obviously not equal to the UK's, but it's not as if they stopped producing dreadnoughts. Germany
wanted to be able to concede victory in the naval arms race to the UK so that it wouldn't have to spend so much money on battleships (in exchange for British diplomatic concessions to Germany elsewhere) and tried to make an agreement with the UK to that effect; it didn't work, so that didn't happen.