Repeating after the muezzin whilst doing wudoo’ and during Tawaaf

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.