Praise be to Allah.
Ibn Hajar al-Haythami was asked this question, and he replied as follows:
When a person is doing wudoo’, he should respond, because the Sunnah for the one who is doing wudoo’ is to refrain from speaking apart from making dhikr (remembering Allaah). There is some dispute as to whether making specific dhikr for each part of the body washed is mandoob (recommended) or not, but the more correct view is that it is not mandoob, as al-Nawawi said, because the ahaadeeth concerning that are not free of lies and narrators who were accused of lying. (In other words, the specific du’aa’s whilst washing the various parts of the body during wudoo’ are not saheeh).
With regard to responding (to the muezzin), it is mandoob according to scholarly consensus, hence they said that it is also mandoob for one who is doing Tawaaf, even though he also has others dhikrs which he should recite according to scholarly consensus, so that is more appropriate in the case of the one who is doing wudoo’.
With regard to finishing wudoo’, if a person’s finishing wudoo’ coincides with the muezzin’s finishing the adhaan, then he may recite the dhikr of wudoo, as stated by al-Balqeeni in his fatwa, giving it precedence over the dhikr to be recited immediately after the adhaan, because the dhikr for wudoo’ has to do with the act of worship that he has just completed, then he may recite the dhikr immediately following the adhaan.
He said, it is good if he recites the Shahaadatayn of wudoo’ then the du’aa’ following the adhaan, because it has to do with praying for the Prophet (peace and blessings of Allaah be upon him), then he may make du’aa’ for himself.