Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
If, while taking a shower or bath, you washed/gargled your mouth and rinsed your nostrils, then you need not to make a fresh wudu.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
If, while taking a shower or bath, you washed/gargled your mouth and rinsed your nostrils, then you need not to make a fresh wudu. Your wudu is already included in your shower or bathing if you had formulated the intention for the same.
If, however, you did not wash your mouth or rinse your nostrils while having your shower or bath, then you should make wudu. The reason for this is that washing the mouth and rinsing the nostrils is part of wudu.
Therefore, in case of omitting the same, one should make wudu. Likewise, one is advised to make wudu if after having had a shower or a bath he/she has intentionally touched the private parts.
Moreover, the late Sheikh Sayyed Sabiq (may Allah bless his soul) states the following in his well-known book, Fiqh As-Sunnah:
If a person performed post-sex ghusl but did not make ablution, the ghusl will suffice.
Aishah (may Allah be pleased with her) said, “The Messenger of Allah did not perform ablution after ghusl.”
Ibn Umar (may Allah be pleased with him) said to a man who had told him that he performed ablution after ghusl, “You went too far.”
Says Abu Bakr Ibn Al-Arabi, “There is no difference of opinion among the scholars that ablution falls under the category of ghusl.
If the intention was to remove sexual impurity, it also includes the minor impurities, as what sexual impurity prevents is greater than what the minor impurities prevent. The smaller one falls under the greater one, and the intention for the greater one suffices.
Almighty Allah knows best.