The Messenger of Allah ﷺ would perform the ritual bath, then pray the two units of prayer followed by the dawn prayer, and I do not recall him renewing his ablution after the bath.
حدثنا عبد الله بن محمد النفيلي حدثنا زهير حدثنا أبو إسحق عن الأسود
كان رسول الله ﷺ يغتسل ويصلي الركعتين وصلاة الغداة ولا أراه يحدث وضوءا بعد الغسل
[Its chain is daif]. Ahmad narrated it in 6/119 from the hadith of Zuhayr bin Muawiya, through this route. Tirmidhi narrated it, no. 107, and Ibn Majah, no. 579, in abridged form. Tirmidhi said, "hasan sahih." Hakim graded it sahih according to the criteria of the two Shaykhs, 1/153, and Dhahabi agreed with him. The hadith has corroborating witnesses.Abu Ishaq did not explicitly state that he heard this particular wording directly.