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 Islam Q & A, states:
Imam Ahmad was of the view that it is obligatory, and he quoted as evidence the hadith narrated from the Prophet (peace be upon him), according to which he said: “There is no wudu for the one who does not mention the name of Allah.” (At-Tirmidhi)
The majority of scholars – including Imam Abu Hanifah, Imam Malik and Imam Ash-Shafa`i – and according to one view narrated from Imam Ahmad, are of the view that saying Bismillah is one of the Sunnahs of wudu, and that it is not obligatory.
They quoted evidence for its not being obligatory, such as:
The Prophet taught a man how to do wudu, and said to him: “Do wudu as Allah has commanded you.” (At-Tirmidhi) This is referring to the words of Allah: “O you who believe! When you intend to offer the prayer, wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles.” (Al-Ma’idah 5:6)
And there is no mention of saying Bismillah in what Allah commands here. (Al-Majmu` by An-Nawawi, 1/346)
Abu Dawood narrated a more complete version of this hadith, which more clearly demonstrates that it is not obligatory to say Bismillah when doing wudu.
It was narrated from Abu Hurairah that the Messenger of Allah (peace be upon him) said: The prayer of any one of you is not complete unless he does wudu properly as enjoined by Allah. So he should wash his face and his hands up to the elbows, and wipe his head, and wash his feet up to the ankles…”
The Prophet did not mention saying Bismillah, which indicates that it is not obligatory. (Al-Sunan al-Kubra by Al-Bayhaqi, 1/44)
Many of those who described the wudu of the Prophet did not mention that he said Bismillah; if it was obligatory it would have been mentioned.(Al-Sharh al-Mumti`, 1/130)
This is the view favoured by many of the Hanbalis such as Al-Khuraqi and Ibn Qudamah. (Al-Mughni, 1/145; al-Insaf, 1/128)
Among contemporary scholars, this view is favoured by Sheikh Muhammad ibn Ibrahim and Muhammad ibn `Uthaymeen (may Allah have mercy on them both). (Fatawa al-Shaykh Muhammad ibn Ibraihim, 2/39; al-Sharh al-Mumti`, 1/130, 300)
They responded in two ways to the hadith which is quoted by those who say that saying Bismillah is obligatory:
1 – That the hadith is weak.
It was classed as weak by a number of scholars including Imam Ahmad, Al-Bayhaqi, An-Nawawi and al-Bazzar.
Imam Ahmad was asked about saying Bismillah when doing wudu, and he said: There is no sound hadith concerning that, and I do not know of any hadith that has a good isnad to support that. (Al-Mughni, 1/145)
2– If the hadith is sound, then what it means is “no perfect wudu” and not “no valid wudu”. (Al-Majmu’, 1/347; al-Mughni, 1/146)
Based on this, then the hadith – if it is authentic – indicates that it is mustahabb to say Bismillah, but it is not obligatory.
Based on this, if a Muslim does wudu and does not say Bismillah, then his wudu is valid, but he is missing out on the reward for doing this Sunnah. To be on the safe side, the Muslim should not omit to say Bismillah when doing wudu.
Almighty Allah knows best.