Bathing on Rosh Hashana
It is forbidden to heat up water on the festival to wash one’s body, since it is not a daily need (not everybody washes his whole body every day). However, one may warm water to wash one’s face, hands and feet only (511:2).
If the water was heated up from the festival eve, then Sephardim may wash their whole body in it. One can also wash oneself with water heated by solar heating, because the sages only forbad heating up water on the festival itself, but any water that was heated up before the festival or heated up by itself, one may wash himself with them. (Chazon Ovadia 511:2)
Bath or public bath
The permission to wash with hot water applies when one takes a bath in one’s own home, but one who goes to a joint public bath like a mikvah, may not wash even his face, hands and feet with hot water. This is because in the past, wicked bath attendants heated water on the festival against the law while saying that they heated it from the festival eve. However, one may wash up with cold water.
AshkenazimThe custom of the Ashkenazim is to be stringent and not wash their whole body at once, even with water that was heated up on the festival eve. However, one can be lenient and wash each limb separately. At any event, it is permitted to wash a baby in hot water even on Shabbat (Rema 511:2 and Mishna Berura ibid.)