The British Government offered asylum initially, but the British King George V (Nicholas' cousin) privately argued against the idea. Although constitutionally it was Parliament's decision George V had enough influence to ensure it didn't come to fruition.

George V's purported reasoning for abandoning his cousin was that he was perceived in Russia as being a tyrannical autocrat and that his wife was seen as pro-German.

http://www.bbc.co.uk/history/british/britain_wwone/cousins_at_war_01.shtml#five

According to Miranda Carter in George, Nicholas, and Wilhelm, Kaiser Wilhelm II, who was also a first cousin of Nicholas II, claimed in his memoirs that he tried to secure asylum for the Romanovs. However, Wilhelm never explained why his attempt failed and there are enough fabrications in his memoirs that it is possible it never happened or was exaggerated.