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:
Answering your question, The Fatwa Center at Islamweb, states:
Saying ‘Bismillah’ (In the Name of Allah) when starting wudu (ablution) is desirable but not an obligation according to the view of the majority of the scholars and if one does not say it, his wudu is valid.
The view we adopt is that if a person performs wudu inside the toilet, then the most preponderant opinion of the scholars is that he should not say the mention of Allah that is said when performing wudu out of glorifying the mention of Allah.
Therefore, it is disliked for a person who is inside the toilet to mention Allah even for performing wudu.
Whoever wants to perform wudu inside the toilet, then he can mention the Name of Allah immediately before entering, and then he enters and performs wudu. This is more appropriate to do than mentioning Allah inside the toilet.
However, some scholars permitted mentioning Allah when performing wudu in the toilet. For instance, Ibn `Uthaymeen said:
“If a person wants to perform ablution inside the toilet, he should say ‘Bismillah’ even inside the toilet and he is not sinful for this because the word ‘Bismillah’ is not a Quran; rather, it is a kind of mention of Allah, but if he says it in his heart and not with his tongue, then this is good, and if he does not say it either by his heart or with his tongue, he is not sinful.”
He also said:
“Saying ‘Bismillah’ in the toilet and the like is permissible because most of the toilets that we have nowadays are clean because water removes the impurity and takes it away, and if one wants to say ‘Bismillah’ in his heart only without uttering it with his tongue, then this is good.”
Almighty Allah knows best.