This is a most interesting question and many historians have explored to some degree or another the exact influence of the whole "family ties" monarchy that existed in Europe at the time. I'll attempt to give a somewhat condensed but also fairly lengthy answer on the question, as the period is definitely one of the most heavily researched and debated in modern historiography. The obligatory "too long didn't read answer" is as follows: Wilhelm was the "runt of the litter" and King George V was afraid of antagonising the Provisional Government in Russia.

Onto the longer explanations. While you are indeed correct in saying that Kaiser Wilhelm II, Tsar Nicholas II and King George V were first cousins, to be incredibly pedantic they were also technically fifth cousins as all three were related to King George II. Regardless of how picky you might want to be on the degree to which all three of them were related, it's often accepted in historiographical circles that Wilhelm II was the so-called "runt of the litter". Despite the fact that Queen Victoria may (see the quote below from the records of her relations)cherished "little Willy" the most out her entire extended family, the same certainly wasn't true between Wilhelm II and his two other cousins.

"a clever, dear, good little child, the great favourite [Wilhelm II] of my beloved Vicky".

Case in point: photographs. The sheer number of photos containing just King George V and Tsar Nicholas II portray how incredibly close these two were: they were practically twin brothers (quite literally, they looked almost identical). Kaiser Wilhelm however, was a bit worse off with his weak physique and limp arm, he wasn't particularly well-received by his fellow monarchs and so from the start of his reign over the German Empire he was mostly antagonistic towards both of them. Keep in mind that it was he who instigated the Naval Arms Race of 1912 against the Royal Navy, and it was he who issued the "Blank Cheque" with Austria-Hungary in the July Crisis of 1914 that many historians have blamed as a major "domino" in the First World War's escalation.

Now of course, that's not to say that "Nicky", "George" and "Willy" were not close before the war began. In fact, quite the opposite: Tsar Nicholas II and Kaiser Wilhelm II maintained informal communications through the Willy-Nicky Telegrams right up until the German-Russian declarations of war. Below is one such telegram, sent on July 29th 1914 (just a few days before Germany joined Austria-Hungary against Serbia):

I foresee that very soon I shall be overwhelmed by the pressure forced upon me and be forced to take extreme measures which will lead to war. To try and avoid such a calamity as a European war I beg you in the name of our old friendship to do what you can to stop your allies from going too far. Nicky.

Of course, Kaiser Wilhelm (and all three monarchs for that matter) had to attend to their own duties as sovereign leaders before considering their family ties. In this case, the Kaiser had to honour the conditions of the Central Powers Alliance and back his Austro-Hungarian counterpart in threatening military action over Serbian aggression. Likewise, Tsar Nicholas II felt compelled to protect his Slavic brethren (Serbia) and this came before his family loyalty. King George V did not make as major an attempt to appeal to family bonds before Britain joined in the war, but he most certainly did not also want to find his nation at war with Germany (the two surprisingly shared more in common than you might think!). Here's an excerpt from Margaret MacMillan's The Rhyme of History on just how simultaneously cordial yet "semi-hostile" the two powers were at the outbreak of war:

One hundred years ago, on the eve of World War I, Britain, the world’s greatest naval power, and Germany, the world’s greatest land power, were each other’s largest trading partners. British children played with toys, including lead soldiers, made in Germany, and Covent Garden resounded with the voices of German singers performing German operas. Moreover, the two nations shared a religion—the majority in both was Protestant—and family ties, right up to their respective monarchs. But all that did not translate into friendship.

So in the end, what it came down to was national duty and the opinions of the monarchs' advisors (or in King George V's case, Westminster). Now onto part two of your question: why did King George abandon his closest relation (or at least in terms of family ties) during Tsar Nicholas' darkest hour?

The answer once again lies in national duty over family loyalties: Britain did not want to antagonise the new Provisional Government in Russia. The Romanov Dynasty (which Tsar Nicholas II was the last ruling member of) had been deposed in the 1917 revolution, and Britain had formally endorsed the democratic Provisional Government under Alexander Kerensky. Whilst King George V wanted to provide asylum to his beloved Nicky, he could not in all conscience do so without betraying the spirit of both his British public and the ruling party in Parliament. In fact, it was his own secretary Lord StamfordHam who finally persuaded the king to "let go" of his former closest friend and family. As historian Helen Rappaport, remarks in The Race to Save the Romanovs: The Truth Behind the Secret Plans to Rescue the Russian Imperial Family:

“[Not to] exonerate King George, not at all. He had a role he could have played more effectively…But the chips were down and they had to choose.”

The "chips" in this case was quite simply nothing less than the entire international relations situation between Russia and Britain following the carnage of the Russian Revolution and not to mention the ongoing Great War on the Continent. Britain needed to keep Russia in the war until American troops could manage to ship their way across the Atlantic and reinforce the Western Front (of course, with the October Revolution and the Bolsheviks in power Russia's engagement in WWI ended very shortly thereafter).

So there we have it then: those these three monarchs were indeed cousins from the same family tree and could all trace their lineage back to "dear old Queen Vicky", they were monarchs of a nation first and foremost. Their family ties, though appealed to in the waning days of peace before the storm of the First World War, were not enough to save their countries from mobilising and marching off into the trenches of the War to End All Wars. As for why Tsar Nicholas II and his family was left to die at the hands of the Bolsheviks, the British were in no mood to be seen as monarch harbouring backstabbers of the Kerensky government, and even King George V's brotherly relation to his cousin could not save the Romanovs. Hope this helps answer your question and shed some light on the topic! Fascinating read by the way!