This book, "Nicholas and Alexandra" by Robert K. Massie, was written decades ago, but it's an interesting start - sorry, no page numbers because I'm using an ebook:

To begin with, one passage in Rasputin's telegram - "Do not allow the doctors to bother him too much" - was excellent medical advice. With four doctors (...) taking his temperature, probing his leg and groin, Alexis was probably denied the total absence of trauma he desperately needed. A clot, gradually formed, still fragile, could easily have been dislodged in the course of one of the doctors' frequent examinations. When at last they left Alexis alone, either because they had given up or because of Rasputin's advice to the Empress, the effect could only have been good.

Massie goes on to theorise that the calmer emotional state of his mother, soothed by Rasputin's intervention, may have contributed.

Alone, the new aura of calm and confidence probably could not have stopped the hemorrhage. But together with the natural reduction of the loss of blood caused by lowered blood pressure and the slow formation of clots, it may have helped. It could even, as Alexandra believed, have been the factor which turned back the tide of death.

Somewhere else, I read that, being ordered to leave Alexis alone, the doctors also stopped giving him aspirin, and since aspirin is a blood-thinner, it's obvious that stopping that was a good idea.

The book was written when the genome was still much of a mystery and many documents were still in Soviet archives, presumably inaccessible to Massie, so there may be a more up-to-date explanation. In fact, Massie himself wrote an update after the fall of the Soviet Union, but I haven't read it and don't know if it includes new information about Alexis' cure. Whatever his status as an historiographer (lack of distance from the historical figures), his book is an entertaining read if you're interested in Russian history.