Allaah hasnot prescribed any prayer at noon time on Fridays except one obligatory prayer for men who are residents and are settled (i.e., not traveling), free and accountable, which is Jumu’ah prayer. If the Muslims do that, then there is no other obligation upon them, be it Zuhr or any other prayer.

There is a town where there are nearly thirty-five mosques in which
Jumu'ah prayers are held. When the worshippers finish Jumu'ah, they
pray Zuhr afterwards. Is this actionpermitted or not?
Praise be to Allaah.
It is a well established fact in Islam (such that no Muslim has any
excuse for not knowing),and the shar'i evidence indicates, that Allaah
hasnot prescribed any prayer at noon time on Fridays except one
obligatory prayer for men who are residents and are settled (i.e., not
traveling), free and accountable, which is Jumu'ah prayer. If the
Muslims do that, then there is no other obligation upon them, be it
Zuhr or any other prayer. Jumu'ah prayer is the obligation of that
time. The Prophet (peaceand blessings of Allaah be upon him), his
companions (may Allaah be pleased with them), and the righteous salaf
who came after them, did not pray any other obligatory prayer after
Jumu'ah. The action to which you refer was introduced many centuries
later, and undoubtedly it is one of the newly-invented innovations of
which theProphet (peace and blessings of Allaah be upon him) said:
"Bewareof newly-invented matters, for every newly-invented matter is
an innovation, and every innovation is a going astray." (Narrated by
Abu Dawood, 3991)
And he (peace and blessings of Allaah be upon him) said: "Whoever
innovates something in this matterof ours [Islam] that is notpart of
it will have it rejected." (Narrated by al-Bukhaari and
Muslim).Undoubtedly praying Zuhr after Jumu'ah is something innovated
which is not a part of the matter of the Prophet (peace and blessings
of Allaah be upon him), so it is to be rejected and is to be included
in the innovations and misguidance against which the Prophet (peace
and blessings of Allaah be upon him) warned us. The scholars have
drawn attention to that, including Shaykh Jamaal al-Deen al-Qaasimi,
in his book Islaah al-Masaajid min al-Bida' wa'l-'Awaa'id and Shaykh
Muhammad Ahmad 'Abd al-Salaam inhis book al-Sunan wa'l-Mubtada'aat.
If someone were to say, "We only do that in order to be on the safe
side, lest our Jumu'ah prayer was not valid," our answer is that he
should be told that the basic principle is that theJumu'ah prayer is
valid and is sound and that it is not obligatory to pray Zuhr, indeed
it is not permissible to do so at the time of Jumu'ah for those for
whom Jumu'ahis obligatory. Being on the safe side is prescribed in
cases where the way of the Sunnah is not clear, and there is some
doubt and uncertainty. But in a case such as this it is not the matter
of doubt; rather we know from the evidence that what is obligatory is
Jumu'ah prayer only, so it is not permissible to do something else
instead or in addition to it on the grounds that this is an action
with the intention of being on the safe side, or to institute a new
way which Allaah has not prescribed. Praying Zuhrat this time is
contrary tothe shar'i evidence which is well establishedand no Muslim
has any excuse for not knowing, so we must not do this and we must
beware of it. There are no grounds for doing this, rather it comes
from the waswaas (insinuating whispers) of the Shaytaan which he
whispers to people so as to prevent them from following true guidance
and institutes for them areligion which Allaah has not allowed. This
is similar to the way in which he makes the ideaof "being on the safe
side" when doing wudoo' attractive to them, until he makes wudoo' a
torment for them, and it takes all their time; whenever they have
almost finished, he whispers to them that they have notdone it
properly or they have not done such and such. He does the same thing
to some of them inthe prayer; when a person says takbeer for the
prayer, he whispers to him that he did not say the takbeer, and he
keeps whispering to himthat he did not say takbeer, and the person
keeps on saying takbeer after takbeer, until the first rak'ah is over,
or until the recitation or most of it is over. This is one of the
traps and snares of the Shaytaan, who is keen to annul thegood deeds
of the Muslim and make him confused about his religion. We ask Allaah
to keep us and all the Muslims safe and sound from his traps, snares
and insinuating whispers, for He is the All-Hearing, Ever-Near.
Conclusion:
Praying Zuhr after Jumu'ah is a bid'ah and a going astray, instituting
a new way which Allaah has not prescribed. We must not do this; we
must bewareof it and warn people against it, and limit ourselves to
doing Jumu'ah prayer only, as was the way of the Messenger of Allaah
(peace and blessings of Allaah be upon him), his companions after him
and those who have followed them in truth until the present day. This
is the truth concerning which there is no doubt. Imaam Maalik ibn Anas
(may Allaah have mercy on him) said: "The later generations of this
ummah will not be reformed by anything except that by which thefirst
generation was reformed." This was also the view of other imaams after
him. And Allaah is the source of strength.
Majmoo' Fataawa wa Maqaalaat Mutanawwi'ah li Samaahat al-Shaykh
al-'Allaamah 'Abd al-'Azeez ibn 'Abd-Allaah ibn Baaz (may Allaah have
mercy on him), part12, p. 363