Oh, finally something I can answer! There was, in fact, such as movement. They were called "Pharaonists" and reacted towards both the increasing role of Islam in society and the growing popularity of Pan-Arabism (although Pan-Arabism as defined by Nasser came to be more about Egypt than an Arab unity) in the first half of the 20th century. They also contributed to the popularization of ancient Egyptian culture and civilization of the 19th and 20th century.

In fact, the "Arab identity" of Egypt is quite recent, and "Arab" was as recent as the 1920s more often used to describe the bedouins of Sinai than the cosmopolites of Cairo. The influential and popular nationalist Sa'ad Zaghloul, who founded one of the most succesful political parties in pre-independence Egypt, the Wafd party, was very critical of Arabism, as was the important thinkers and writers Taha Hussein, Muhammad Abduh' and Ahmad Lutfi al-Sayyid. Indeed, Egyptian nationalism prior to the revolution of 1952 was just that: Egyptian, not Arab.

Some scholars, such as Nadav Safran, former director of Middle Eastern studies at Harvard University, even argue that political pan-Arabism was rooted religiously, not in the ideas of Al-Afghani, ‘Abduh, or Al-Banna, but in the aspirations of King Fuad I (r. 1922-1936) to unite the Arab nation under an Egyptian caliphate.

In an article soon to be published I argue:

The general consensus, however, tends to be that the Wafd party, and especially Zaghlul, opposed pan-Arabism in favor of a secular, Egyptian nationalism. This opposition of the Wafd party, as Safran mentioned, changed during the 1940s. The influence of king Fuad, and later his son Faruq (r. 1936-1952), may have played a part in that transition, but in all likelihood the increased tensions in the neighboring British Mandate of Palestine, between Jewish settlers and Palestinians, led to a reactionary Arab nationalism. This ideological transition was, of course, not limited to the Royal Palace or the Wafd party, and may very well have grown stronger in the Egyptian Army, as they were sent to fight the newly formed state of Israel in 1948. From the viewpoint of the Palace, the reason for going to war was less to support their brethren and more to hinder the king of Transjordan, Amir Abdallah (r. 1946-1951), of winning against Israel and then annexing Palestine to his own kingdom. But nevertheless, the army fought in a war that was largely justified as an “Arab cause”, and it was indeed a popular war, in part thanks to the Muslim Brotherhood having called for a jihad against Israel. Even though the war was a humiliating defeat, the strong nationalism that had been fostered in the army had now been both widened and strengthened to include all of the Arab Nation.

Sources and further reading:

Fouad Ajami, "The End of Pan-Arabism," Foreign Affairs (1978).

Anwar G. Chejne, "Egyptian Attitudes toward Pan-Arabism," Middle East Journal (1957)

Albert Hourani, Arabic Thought in the Liberal Age: 1798-1939. Cambridge: Cambridge University Press, 1983.

Nadav Safran, Egypt in Search of Political Community: an Analysis of the Intellectual and Political Evolution of Egypt, 1804-1952. Cambridge: Harvard University Press, 1961,