Below is Shaikh al-Albani’s position on the issue… *& there is a difference of opinion regarding praying duhr or not*
Questioner: Also a question out teacher about the Jummah prayer if it comes on the day of Eid, does it drop altogether? That is the permission that a Muslim does not have to pray it?
Questioner: He does not pray in its place anything: that is: Dhuhr for example?
Shaikh: No, verily the Imam must pray Jummah
Questioner: He prays Jummah?
Shaikh: The Imam
Shaikh: Those who pray Eid, then they can choose. Whoever wants can leave it off and whoever wants can pray
Questioner: Does he pray dhuhr if he did not pray Jummah?
Shaikh: No, if the dhuhr prayer falls as a pardon if the Jummah prayer falls, then therefore the dhuhr prayer falls as well.
Questioner: Jazakallahu Khairan
Shaikh: Wa iyyaka
[Silsilatul Huda wan-Nur no. 29]
Faisal Ibn Abdul Qaadir Ibn Hassan