I believe that would be the film, Nicholas and Alexandra, rather than Robert Massie's book, depicting Alexei giving his father a very quick hug.
We only can go by what was observed by actual witnesses. And that would be, as far as we know, the assassins, plus (if I recall correctly, from FOTR), a guard who was peering through one of the room's windows. (The basement being only partially subterranean.)
What was reported by these sources was that Alexei was frozen with fear. However, given Alexei's love for his father, it was not unreasonable--in my opinion--for the Nicholas and Alexandra screenwriters to have him quickly hugging his father. (This is why I generally prefer scripted programming to the new, so-called "reality" shows . . . Truth can be portrayed through fiction, whereas reality shows are so often . . . well, banal!)