When Eid and Jum’ah coincide, then Eid Salah is Waajib, whilst Jum’ah Salah in Fardh. It is not permissible for those residing in towns to leave any of [the two prayers]. The villagers, whom upon them Jum’ah and Eid are not Wajib anyway, are given leeway to return to their villages after praying Eid Salah [in the city], it is not upon them to pray the Jum’ah Salah. This is because, if the villagers enter the city, and do not remain until Zawaal time, Jum’ah will not be Fardh upon them. [For them] returning [to their villages] before Zawaal is permissible.

[Imdaad ul-Ahkam – Mufti Zafar Ahmad Al-Thanwi 1/779]

Evidence:
1) Abu Ubayd said: I witnessed Eid Salah with Uthman Ibn Affan. He arrived, prayer, then turned towards us. He delivered the sermon and said: “Indeed, two Eids (Eid & Jum’ah) have come together for you on this day of yours. Whoever wants to wait for the Jum’ah, from amongst the villagers, he may wait. Whoever wants to return, then I have granted him permission.” [Muwatta Malik]
Mufti Zafar Ahmad:
“Uthman had given a concession during Jum’ah for the villagers because they are not from the city. This is the opinion of Abu Hanifah. Uthman had said this in front of the presence of Sahabah. If this concession included the villagers and those of the city together, just as Ahmad Ibn Hanbal (May Allah have mercy on him) considered, they [Sahabah] would have disputed him, due to the specification of [the concession] with the villagers. Thus, it is established that the concession is specific with the one whom upon him Jum’ah is not necessary. Therefore, Jum’ah will not be left due to Eid.
…
[As for the Hadith of Ibn Majah/Abu Dawud] if we accept its authenticity, in attribution to Rasulullah صلى الله عليه وسلم, then we say: The villagers would gather for the two Eid Salah, they would not gather for other than them [two Eid Salah], just as the norm. Their wait for the Jum’ah Salah, after the completion of Eid was difficult for them. So, when Rasulullah صلى الله عليه وسلم would finish from the Eid Salah, he summoned his announcer to call out: “Whoever from amongst you intends to pray Jum’ah, then he may pray. Whoever intends to return, then he may return.” This was addressing the villagers who had gathered there. The indication to this, is that the ‘gathering’ is clear within [the Hadith]. What is indicated by the plural 1st person pronoun [‘We will pray’ in the Hadith of Abu Dawud] is the People of Madinah, without a doubt. There is a clear expression that the addressing is towards the villagers, and not the people of Madinah, with his statement: “Whoever from amongst you intends to pray.”
[SEE: ‘Ila Ul-Sunan 8/92-94]