Constantinople
I think the British Government would have seen giving refuge in one of its dominions in the same way as giving refuge in Britain, with the added element that the dominion government would need to agree.
As I understand it, in WW2 Queen Wilhelmina stayed in Britain (along with Prince Bernhard). It was the then Crown Princess Juliana who went to Canada, along with her children, primarily to safeguard the Dutch succession. Wilhelmina (and, for example, Haakon of Norway, who also stayed in Britain) was in a rather different position from Nicholas, being the head of state of a democracy under Nazi occupation. Both Wilhelmina and Haakon, if little known here before the war, soon acquired personal popularity as a result of their unpretentious natures and obvious devotion to their respective countries.
As Alixz has said, in 1917, life was cheap compared with today, and in April 1917, with Russia apparently settling down under the Provisional Government, there was probably no sense of an immediate threat to Nicholas and family. Although we can now draw comparisons with Louis XVI and family, where the threat took some time to develop, that was something nobody knew then. Further, April 1917 was a low point in the war (huge losses on the Somme, the resumption of unrestricted submarine warfare, the Americans now in the war but not yet in a position to do much). The safety of the 'tyrant' Nicholas was not going to be high on the British priority list.
Ann