I can do my best to answer your question, but since my study only very briefly touched upon Iran and the Cultural Revolution I probably can't give you as much information as you want.

To first understand why such a change could happen, it is a good idea to know the background. The Shah of Iran was not a very well-liked man. His secret police, the SAVAK, was extremely effective in rooting out, torturing and executing opposition and his reliance on the west to stay in power made him very unpopular among the Iranian population.

During this time of oppression, which in 1975 banned the only other political party, making it a one-party state, the only real way to express dissent was through religion. Muhammad Reza Shah did his best to secularize the country and bring it back to its more Persian roots, but due to the power of the Ulama (guardians of the legal tradition of Islam), this was not successful.

Thanks to Ayatollah Khomeini's anti-shah preaching during his time in exile, he was a reasonably popular man within the Iranian opposition, and Khomeini's most important base of support was his claim that the Shah was a puppet of the US, but more importantly, not following the true path of Islam. For most people then, Islam became the way they could strengthen their nationalism, evict western powers from their country and end the oppression of the Shah. This popular appeal was the reason Khomeini came into power.

And here comes the answer to your question: Khomeini divided the government in two, one part was headed by the leader of the by Iranian Freedom Movement, Prime Minister Mehdi Bazargan (put in power by Khomeini), and the second part was the Council of the Islamic Republic. This second group was composed of mainly Ulama and headed by Khomeini. This Council made it impossible for Bazargan to pass any secular legislation. Here the people in power weren't necessarily replaced, but rather just overshadowed by their Islamic counterparts.

Specific examples of what you ask for in your question can be found in the legal system. Here, western courts were dissolved and instead introduced courts where the judge needed a certain degree of competence in Islamic law. Any former judge who did not have this was removed.

A small sidenote is important though: due to the nature of governments where people aren't elected, having the right connections is very important. This means that if someone like your uncle is high enough within the religious government to put people into work positions, he might put a religious family member in said position instead of a more qualified individual, simply because he can.

Sources: A History of the Modern Middle East by William L. Cleveland and university lectures