Two things:
I think most of the blame for the decision not to extract the Imperial Family has to ultimately rest with the British Government, and not with the Royal Family. A decision to offer Nicholas a refuge within the British Empire couldn't have been made by any King without the approval, and indeed the instigation, of the Prime Minister and Cabinet.
That being said, it defies belief that George V did not realize the potential danger in which NAOTMAA found themselves in after March, 1917. Alexander II was assassinated; there were numerous murders (Grand Duke Serge and Stolypin, to name only two) prior to the actual overthrow of the monarchy, and no reason to suppose that there would be a non-violent transfer of power. In fact, by the time the Bolsheviks seized control it must have been obvious in what danger the Imperial Family was placed. As for the idea that he may have thought they would be allowed to go quietly into exile, neither the Austrian or German Kaisers was overthrown until after the Tsar, and Wilhelm II was prudent enough to abdicate near enough to the Dutch border to get across it. Had he been imprisoned in Berlin at the end of the war, it is likely he would have been either extradited to the Allies as a war criminal, or possibly executed by his own former subjects during the Bolshevik uprisings at the end of the war.
I don't think you had to be too sensitive to realize that the former absolute ruler of the Russian Empire, a prime target of assassins throughout his entire reign, was going to be in danger when he was deposed.