According to Marina Botkin Schweitzer (Gleb Botkin's daughter) Tatiana Botkina was already with her father in Tobolsk when Gleb joined them in the fall of 1917, and the children remained in Tobolsk throughout the Imperial Family's captivity there. So, yes, Tatiana Botkina had direct, daily contact with her father at the time of Aleksei's illness. Whether or not she told the truth, embellished the story, or just plain made up the cause of the hemorrhage is another matter...
However, I was incorrect in saying that Gleb was writing stories for the tsesarevich in the spring of 1918 -- Gleb's stories were written for Aleksei before the October Revolution of 1917.