What if....

That makes sense to me, TampaBay. Royals are picky that way, I guess. Everyone has to be of an equal royal status or have an "English" last name.  Mountbatten is the English version of the surname, Battenburg, which was his mother's surname. (His mother Princess Alice, was a great-granddaughter of Queen Victoria.)

Actually, no.  The children of Elizabeth II and their descendants derive their rank from their grandfather, George VI who issued the English equivalent of a ukase (sorry, I can't think of the correct term just now, Letters Patent? maybe) endowing EII's children with the HRH and princely rank.  This happened during her first pregnancy (Charles) and was meant to prevent him from being born without rank.  Since the sovereign is the "fount of honour" in England, I assume that had George VI not done so, Elizabeth II by virtue of her sovereign status could have done so in her own right after she ascended the throne.  But it wasn't necessary.  Pax, Nadezhda.

Nikolai would (apologies about the harshness of this statement) have more than likely outlived Alexei, because of the nature of Alexei's medical condition, (which was a State secret).

Alexei's illness was not a state secret.  The main Alexander Palace website contains a newspaper article from 1912 reporting the history of hemophilia in Victoria's progeny and specifically mentioning Alexei as a sufferer.  (There is a lengthy thread on this Forum discussing just how widespread the knowledge was that Alexandra was a likely carrier even before her marriage and how common was the knowledge -- at least among the elite classes inside and outside of Russia -- of Alexei's hemophilia.)

I have no idea how the notion came about that Alexei's illness was a closely-guarded secret at the time, although the notion is certainly widely accepted these days.  My only guess is that no one dared speak openly about the illness during Nicholas' reign, because of his and Alexandra's determination to treat Alexei as heir -- and that reticence has subsequently been taken as lack of awareness.