Going back to what Greenowl had to say about age of marriage for royal men, I think it unlikely that Alexei would have married under 20. I would not claim my list to be complete, but I think that by the early 20th century the 'typical' marrying age for princes was 25-30 (for example, George V was 28, his brother Eddy due to marry at 28, Nicholas II 26, Henry of Prussia 26, Ernst Ludwig of Hesse 26, Arthur of Connaught (snr) 29, Arthur of Connaught (jnr) 30). Some did marry at 22-23, but two youngest bridegrooms I've found on a fairly quick scrutiny were Alfonso XIII of Spain, who was two weeks past his twentieth birthday, and Charles Edward of Coburg, who was 21. Both of them had particular need to produce heirs, especially Alfonso, who had been born posthumously, had neither brothers nor paternal uncles and ruled a country where female succession had not been fully accepted.
On that basis, Alexei might have married at 20-21, but finding a suitable bride would have been difficult. Was it recognised at the time that any daughter of a haemophiliac is inevitably a carrier? Quite apart from the obvious risk that the bride of the Tsarevich might well have found herself the youthful widow of the Tsarevich before very long (I have visions of Alexei sharing his Spanish cousins' love of fast cars!) Given the war, it would have been politically impossible, I think, for him to have married a German princess, and there was also the possibility that female-line descendants of Queen Victoria would have been haemophilia carriers. Then if you have a very young Tsarevich in search of a bride, the age range of suitable girls was going to be narrower then usual.