The Messenger of Allah ﷺ would perform the ritual bath, then pray the two rakats followed by the dawn prayer, and I do not think he would renew his ablution after the bath.
كان رسول الله ﷺ يغتسل ويصلي الركعتين وصلاة الغداة، ولا أراه يحدث وضوءا بعد الغسل.
Narrated by Abu Dawud (250), and this is his wording, and Tirmidhi (107), and Nasai (252), and Ibn Majah (579), all of them from the hadith of Abu Ishaq, from Aswad, from Ayesha, who then related the hadith. In one version, "The Messenger of Allah ﷺ would not perform wudu after ghusl," and in Ibn Majah's version, "after ghusl from janaba." Tirmidhi said, "Hasan sahih." Hakim graded it sahih (1/153), saying, "Sahih according to the standard of the two Shaykhs, though they did not include it." And it is as he said.Tirmidhi said, this is the view of more than one scholar among the companions of the Messenger of Allah ﷺ and the successors, that one should not perform wudu after ghusl.