why did max wait so long to reveal his illness to the family

I think this is because of the circumstances that they are all in. Max is 23 and a strong guy. He does not wish to be a burden to Hans, Rudy or Liesel. They are already risking their lives by hiding him. Max also hopes that his sickness will pass thus he could skip explaining the whole episode. Unfortunately after Liesel brings in the snow to build a snowman in the basement, Max's health worsens.