It wouldn't be possible to claim the Russian throne through one of Nicholas II's daughters. The succession law was semi-Salic, and the effect of it was that females would inherit only when the male line was completely exhausted, and then it would be the female nearest in blood to the last male dynast. So if Tsesarevich Alexei had no children his heir would be his uncle Michael, if he had survived, and if not his father's cousin Kyril or his son Vladimir if Kyril was dead already, not any of his sisters. They would inherit only if Alexei was himself the last male dynast and had no children. Alexei could in any case have well lived to a reasonable age and had a son or sons. His first cousin Prince Waldemar of Prussia was also a haemophiliac and lived to be 56, and although he didn't happen to have children the condition didn't rule it out. Queen Victoria's fourth son Leopold, Duke of Albany, was another sufferer and fathered two children before he died aged 30.
While it is credible that the imperial family could have found refuge in Britain, the Kerensky government was willing to let them go and the British government to offer shelter, the plan failing in part due to the determined opposition of George V, I do not think that political activity by them would have been tolerated. Nicholas II had in any case abdicated, first for himself and then with rather questionable legality for his son, so there could only reasonably be a claim on behalf of the latter
I do not believe though that an imperial government-in-exile would have been allowed to form around him. As already pointed out, sooner or later it would become irrefutably clear that the Soviets had won and Britain would need to establish relations with them, hardly possible while hosting a rival government. I also think it very unlikely that any of the Grand Duchesses would have married anyone near to the British succession, that would have been a considerable no-no on numerous grounds.