I think that to be fair George V's actions are entirley understandable and are not that surprising.
During the First World War George V came under increased scrutiny particularly over his 'foreign' origins although to be fair he was essentially a very English country gentleman, although a bit of a martinet, at heart.
He was also hide bound by the constitutional nature of his crown, something that both Wilhelm and Nicholas mocked, which limited what action he could take.
He was also facing an essentially collapsing Russian position and a government that hastily offered asylum largely to try and keep the provisional government onside in the war.
A government that didn't really consider the long-term impact on the King's position of having his cousin in the country.
Nicholas' reputation in the west had never been particularly good and to the King, who actually had pretty good relations with his left leaning government's in his later reign, housing Nicholas the Bloody would have been a public relations disaster.
George's first duty was the survival of his own country and his own crown not rescuing monarch's who through their own foolishness had found themselves in appalling circumstances.
Nicholas was advised on signing the abdication to leave the country - his mother was told that he should go immediately abroad - he returned to his wife and children perhaps we can understand that but then he also ignored offers to try and get the children out.
By the time the British asylym offer was made (before it was withdrawn at the request of the King) the Petrograd government was already asking that the offer be withdrawn given the Soviet had demanded assurances that the former Tsar would not be allowed to leave.
The window of escape closed before it really opened.
George's ability after that to do anything was pretty hampered as Nicholas really ceased to have any political importance to the British after the fall of the provisional government. George couldn't as many people seem to think simply order his secret service or government to act and nor should he have done.
George V is often the villain of the piece but Nicholas had other cousins including those a damn site nearer who equally did very little to help.
If as you say asylum offer had survived - then how do you get them out - without the Soviet rising against the provisional government whose survival the allies were desperate to ensure - where do they go to - the obvious is Finland then Sweden/Norway/Denmark - but how and what risk of Nicholas in crossing Finland given the only Romanov who was vaguely popular with the Finns was his mother.
When Kerensky fearing for their safety sent them deeper into Russia rather than nearer a friendly border - how is that helping to save them rather than his own crumbling and inept government?
I think George V does indeed bear some blame, if not specifically for the deaths of the Imperial Family, then certainly for the decision not to grant them asylum in England, which asylum would have saved their lives.
History does not generally blame Lloyd George or Parliament for this decision. This is not really disputed even by critics of Lloyd George. George V was not exactly a poor chimneysweep in a mattter as vital as the fate of his own relatives and of what had been a ruling family until a few weeks previously. There is also the evidence of his private secretary (Stanfordham ?) , who wrote of George V's role.
Further confirmation of George V's responsibility may be inferred from the efforts that were made to discourage and pressure his ambassador to Russia , Buchanan not to speak or write publicly about the matter , which would haver made the King look self-centered, callous, and bad.
That the Romanovs were indeed in danger can pretty much be inferred from the very fact that desperate efforts were being made to get them out of Russia. The Provisional Government, especially Pavel Miliukov, knew this and acted accordingly. It wasn't a question of finding the IF a preferred new residence, but rather of finding them asylum, i.e., of saving their lives.
As acknowledged by earlier posters, the St. Petersburg Soviet was intensely hostile to both Nicholas and Alexandra; in fact some delegates openly demanded his execution. The danger to the IF was clear. Even before the Bolsheviks seized power in October of 1917 the SRs's and other left extremists in the Soviet were known proponents of terror ,including against the Romanovs. They had a bad record for it in the previous fifteen years.
And yes, getting the IF to safety in England would have been difficult. But it was by no means impossible. All the more reason for getting the approval of and some zeal shown by the British monarch as well as by the British government for the move. After George V backed away, the window of opportunity was closed and the IF's options were virtually nonexistent and their fate effectively sealed.
Obviously George V didn't know for cetain that the Romanovs ,his cousins, would be executed . But the fact is he could have and should have known that they were in serious danger. The resort of Russian revolutionaries to murder of high government officials and of a Romanov Emperor and a Romanov grand duke was a matter of record. And these were while Romanovs occupied the Russian throne. With the entire Romanov family imprisoned just fifteen miles away from the St. Petersburg Soviet, how could George V not have appreciated the peril and the consequent imperative to offer timely succor?
