The Messenger of Allah ﷺ would perform the ritual bath, then pray two rakahs followed by the dawn prayer, and I do not think he renewed his ablution after the bath.
كان رسول الله ﷺ يغتسل ويصلي ركعتين، وصلاة الغداة، ولا أراه يحدث وضوءا بعد الغسل.
Narrated by Abu Dawud (250), and this is his wording, and Tirmidhi (107), and Bayhaqi (1/179), all of them from the hadith of Abu Ishaq, from Aswad, from Ayesha, who said, and then she mentioned it. In another version with Tirmidhi (107), Ibn Majah (579), and Ahmad (24389), she said, "The Messenger of Allah ﷺ would not perform wudu after ghusl from janaba."Its chain includes Sharik Nakhai, the judge, who has weak memory. Bayhaqi placed a heading over it, saying, "Chapter on omitting wudu after ghusl."I say, these hadiths indicate that ghusl includes wudu and more, and this is why the Prophet ﷺ did not command a man in a state of janaba to perform wudu after ghusl. However, they differed regarding rinsing the mouth and nose, since these remain among the limbs of wudu. Those who held that these two are obligatory said they are indispensable, while those who held that they are not obligatory relied on the hadith of Imran bin Husayn, the hadith of Jubayr bin Mutim, and the hadith of Jabir bin Abdullah mentioned earlier.Shaykh Ibn Uthaymin, may Allah have mercy on him, was asked, "If a person performs ghusl for janaba and does not perform wudu, does this suffice in place of wudu, and may he pray directly?"He answered, may Allah have mercy on him, "Yes, it is permissible for a person in a state of janaba to limit himself to ghusl along with rinsing the mouth and nose, because Allah Almighty said, "And if you are in a state of janaba, then purify yourselves" [Surah Maidah 5:6], and He did not mention wudu. In Sahih Bukhari, from Imran bin Husayn, there is the long story that a man withdrew and did not pray with the people... and the Prophet ﷺ said to him, 'Take this and pour it over yourself'... "He said, "This indicates that when a person performs ghusl from janaba, covering his body with water, it suffices for him in whatever manner it is done. Accordingly, if a person immerses himself in a pool or a pond with the intention of ghusl from janaba, then comes out, rinses his mouth and nose, and goes on to pray, this suffices, because he has purified himself, and rinsed his mouth and nose."