Praise be to Allah.
It is permissible for a person to speak whilst doing his Wudu, because there is nothing to indicate that this is not allowed according to sharee‘ah, and the basic principle with regard to actions is that they are permissible.
It was narrated from some of the scholars (may Allah have mercy on them) that it is makrooh to speak whilst doing Wudu; this is to be understood as meaning that it is preferable not to do that, and it does not mean that it is makrooh in a shar‘i sense. In other words, it is preferable for the person who is doing Wudu not to speak unnecessarily whilst he is doing it.
An-Nawawi (may Allah have mercy on him) spoke of the Sunnahs of Wudu and actions that are mustahabb during it, and he said: He should not speak unnecessarily during it. Al-Qaadi ‘Iyaad narrated in his commentary on Saheeh Muslim that the scholars disliked speaking during Wudu and ghusl. What he quoted about them disliking it is to be understood as meaning that not doing it is preferable. However there is nothing to prove that it is not allowed, so it cannot be called makrooh (disliked) except in the sense that not doing it is preferable."(Al-Majmoo‘, 1/490-491 )
Shaykh Ibn ‘Uthaymeen was asked about speaking whilst doing Wudu: is it makrooh?
He (may Allah have mercy on him) replied: Speaking whilst doing Wudu is not makrooh, but in fact it distracts the individual, because when the one who is doing Wudu washes his face, he should bear in mind that he is obeying the command of Allah, and when he washes his arms, wipes his head and washes his feet, he should bear this intention in mind. But if someone speaks to him, and he speaks to that person, this focus is disrupted and he may also get confused about his Wudu, and he may be faced with waswaas as a result of that. So it is preferable for him not to speak until he has finished his Wudu. But if he does speak, there is no blame on him for that."(Fatawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen).
And Allah knows best.