Is there any situation, bar imminent threat of death, which would allow one to shower with hot water during the Shabbat, using hot water which is heated on the Shabbat, or is it absolutely not allowed?
This article seems to provide a detailed analysis of this question. Excerpt from beginning. (Read the rest of the article as well as cited sources, there):
SUMMARY – on Shabbat one may wash up to half their body in permitted hot water. The types of hot water that is considered permitted to (partially) wash with on Shabbat are, water heated before Shabbat that can be used without causing new cold water to cook; or water that was heated on Shabbat by itself, which includes water from the solar water unit (according to those who rule that solar water heaters are permitted on Shabbat); and possibly water heated up by a non-Jew for his own use. There is a debate about how cold water must be to still be considered "cold" water.