I have a number of interesting books about Queen Mary of England as well as one containing the letters Tsar Nicholas II wrote to the Tsarina Alexandra as well as the daily diaries of the Tsar and his family. The Tsar wrote letters on a daily basis to Alexandra - the letters were in English.
You wrote about the cruel and callous behavior of King George V to his cousin an his family and I hope the above could explain some of the intrigue involved.
By 1890 Queen Victoria was looking for wives for the sons of at the then Edward of Wales - Albert Victor (Duke of Clarence) and George (Duke of York - the later King George V) Both the sons were rather weak characters so Queen Victoria was looking for wives with stronger characters to marry them. Her first effort was to get a wife for Albert Victor - the elder of the two and one of the princesses involved was the future Tsarina Alexandra. She turned the suggestion down flat and so did other princesses of German origin. Queen Victoria then invited Princess Mary Victoria of Teck and her brothers to Balmoral Castle to see whether the Princess will make a suitable future Queen. After that the marriage was arranged and the two got engaged. However, before the marriage could take place Albert Victor died of pneumonia.
The problem at the time was that Princess May of Teck was not a princess of the royal blood and a fairly low candidate to marry Albert Victor since her grandfather - the Crown Prince of Wurttemburg married morganatically a countess from Hungary and renounced the crown of Baden-Wurttemburg. Her father was a favourite of the Kaizer Franz Joseph of Austria and on his request the King of Wurttemburg gave his cousin the title the Duke of Teck. He became involved in the British situation when he married Princess Mary Adelaide of Cambridge. The latter was known as Fat Mary of which the Prime Minmister once wrote that no German Prince would enter into such a vast undertaking. Be it as it may the issue must have affected the later Queen Mary throughout her life - she was by all accounts a hard woman who totally controlled King George V and was a poor unfeeling mother as well.
Even before the death of Albert Victor Queen Victoria wrote to the future Alexandra of Russia to suggest that she marry Prince George - Duke of York. Her answer was that "Prince George was a beloved chum - but she would not marry him", Based on the problems she previously had in finding a wife for Prince Albert Victor Queen Victoria then decided to get Prince George to marry Mary Teck.
There is no way that the later Queen Mary did not dislike the Tsarina with a passion and I believe that is the reason for King George V in refusing to help his cousin and even refused him and his family to flee to England was caused by Queen Mary. Incidentally the king did help the Empress Marie (Minnie) and the Grand Duchess Xenia as well as their associates and friends to escape from Russia and why not the Tsar and his family? They did not even had to live in the UK - they could have settled in any of the Dominions. I think it was entirely due to Queen Mary and the fact that King George V was a weakling under the thumb of his wife.
The above account may give an answer insofar as the actions of King George V in this case is concerned.