The Messenger of Allah (ﷺ) took a bath and offered two rak'ahs of prayer and said the dawn prayer. I do not think he performed ablution afresh after taking a bath.
Sunan Abi Dawud Book 1, Hadith 250
← Previous Hadith The Messenger of Allah (ﷺ) said: If anyone who is sexual defiled leaves a spot equal to the breadth of a hair without washing, such and such an amount of Hell-fire will have to be suffered for it. Ali said: On that account I treated my head (hair) as an enemy, meaning I cut my hair. He used to cut the hair (of his head). May Allah be pleased with him.

Sunan Abi Dawud Book 1, Hadith 249 Next Hadith → one of the Muslims asked, and Zubair reported: Umm Salamah (herself) asked: Messenger of Allah. I am a women who keeps her hair closely plaited; should I undo it when I wash after sexual defilement? He replied (no), it is enough for you to throw three handfuls over it. Then pour water over all your body and will be purified.

Sunan Abi Dawud Book 1, Hadith 251