The point is that doctors' cauterization was not successful. Anna Vyrubova described the event:
"I saw him when he was lying in the nursery: the little one, with waxen face, he had blood-stained cotton in his nostrils."
And of course, the Imperial Pair called Rasputin. The story goes:
''He walked over to the bed, the boy made the sign of the cross... After that, Gregory E. said that nothing was serious and that there was nothing to worry about; then he turned and walked away. The bleeding stopped. The doctors said that they did not understand how it had happened. " (A.Vyrubova).
Pierre Gilliard, who believes in successful doctor' cauterization:
''On the morning of our departure, Thursday, December 16th, Aleksey Nicolaievich, who had caught cold the previous day and was suffering from a heavy catarrh in the head, began to bleed at the, nose as a result of sneezing violently. I summoned Professor Feodorov but he could not entirely stop the bleeding. In spite of this accident we started off, as all preparations had been made for the arrival of the Tsar. During the night the boy got worse. His temperature had gone up and he was getting weaker. ''
'' With infinite care the invalid was taken to the palace. The doctors ultimately succeeded in cauterizing the scar which had formed at the spot where a little blood-vessel had burst.''
I suppose you have read this already, so I can not say how exactly the cauterization had been done -- how about this :
'' The procedure was simple: a piece of metal was heated over fire and applied to the wound. This would cause tissues and blood to heat rapidly to extreme temperatures in turn causing coagulation of the blood thus controlling the bleeding, at the cost of extensive tissue damage. Sometimes liquid nitrogen is used as a less painful alternative, though it is less effective. In the few countries that permit the use of cocaine for medicinal purposes, it is occasionally used topically to make this procedure less uncomfortable; cocaine being the only local anesthetic which also produces vasoconstriction, making it ideal for controlling nosebleeds. More modern treatment uses silver nitrate, a local anesthetic is applied and the procedure is generally painless. The nose may run for up to a week after the procedure. ''