Well, my thoughts were mostly of how Queen V supposed the ladies should manage Eddy, the Queen knew her grandson weaknesses and all, she was fond of Alix but thought very highly of May. And I do not have the idea Eddy was ever really in love with someone unless he was told that someone was somehow available and suitable. So, supposing Alix and Eddy did get married, and he didn't die so soon, do we have any reasons to believe dear Alix would feel more loved by her mother in law Alexandra than she was by her sister in Russia? Do we have any reason to believe she would be less possesive, less shy, less unlucky to have an haemophiliac son?
I am not convinced that Eddy fell in love with women because he was told they were available/suitable. Eddy certainly thought himself he was in love with Alix, who was both available and suitable, and was certainly encouraged by Queen Victoria, but she was very beautiful even for the pretty low standards of current European princesses and given his limited options, there is no reason to believe he wasn't actually smitten. Whether he would have remained in love had Alix been in any way forthcoming is open to doubt, since he in fact subsequently fell in love with Hélène of Orleans who was available but unsuitable, and shortly after that came to nothing he wrote flirtatious letters to Lady Sybil St Clair Erskine who was available/unsuitable (not royal) in which he referred to yet another love who was probably at least unsuitable since he did not name her. There is no indication he fell in love with May of Teck - she was merely Queen Victoria's second choice, another serious young woman who Queen Victoria primarily, but also the Prince of Wales, thought would help to steady this rather feckless young man.
I assume you mean that Alix was shy and unlucky to have a haemophiliac son, and Alexandra was possessive and not likely to be very welcoming to her as a daughter-in-law. Since Alexandra was possessive of her second son George, and would probably have proved to be the same to her first son Eddy (and in reality was difficult as a mother-in-law to Mary), it is likely that she would have been the same as her sister - difficult and possessive - had Alix married Eddy. However, Eddy and Alix would have had longer to develop their roles before taking centre stage as monarchs which of course in Great Britain were not political and there would have been less pressure on Alix as a social leader and her shyness might have been less of an issue.
Since haemophilia is transmitted in the female line, if Alix were to have had sons had she married Eddy, they would have had a high likelihood indeed of having haemophilia. However, there would have been somewhat less pressure on her to have sons since daughters could inherit the throne of Great Britain and that would presumably make the issue less important in a dynastic sense. So I agree with the previous remarks that the things that worked against Alix as Empress in Russia would not necessarily have worked against her in Britain, or at any rate not nearly to the same extent.
The one great problem would have been in WW1 when her Germanic ties would have been quite as disadvantageous in Britain as they were in Russia.
