A few years ago, I was told by a history professor I respect very much that it was the British Petroleum funded coup that indirectly lead to the radicalization of Islam. I can't really remember the details, so I may be missing key points (or I may be completely wrong). His narrative was that, after the coup, all resistance to the government/establishment was shunted into the masques, the only places too sacrosanct for the government to interfere. This caused the radicalism to blend with the religion, giving us what we have today. (I remember him bringing the Sha into the story somehow, but, like I said, I'm really fuzzy on the details -- history has never been my thing, despite plenty of interest and appreciation.)
He even said that, before the coup, it wasn't unusual for women to be allowed out without hijabs and gave a few other examples.
Is the narrative (basically) true? Can anyone fill in the details, or link me somewhere that can?
And was Iran really that liberal? Sources?
While BP and the British Crown were the primary beneficiaries of the coup, most of the funding came from the CIA. In 1952, the CIA spent 10% of its global budget in Iran.
Leading up to the Iranian revolution, there was plenty of opposition from students and other liberal groups, but their leaders were always prime targets for the Shah's SAVAK secret police. As a result, secular opponents of the regime were not nearly as well organized as the mullahs, and their common cause revolution soon fell under the Islamist banner.
Both Shahs of the Pahlavi dynasty pursed a modernist secular agenda similar to that of Kemal Ataturk. Reza Shah Pahlavi banned the chador in the 1930's, and mandated Western garb. He pushed for women to have an education and a role in the workplace.
It's hard to see a direct tie between Iran's revolution and the broader radicalization of Islam. Because of the Shia/Sunni split, Iran was never in a position to export its revolution in a big way.
It could, however, be argued that Iran demonstrated a path to harnessing Islamists as a potent force. This agenda was initially realized in Afghanistan, after the Saur Revolution declared equality of the sexes. President Carter's National Security Advisor Zbigniew Brzezinksi saw an opportunity to bolster the enraged Islamists, prompting the USSR's fraternal invasion. Afghanistan then became a fertile breeding ground for Islamists, supported by Saudi Arabia, Pakistan, and various Gulf states, along with the US. This campaign succeeded so well that the US had to go to considerable lengths to stop the Islamist insurgency from spreading to other muslim-dominated Soviet republics. If you wanted to identify ground-zero for radical Islamism, Afghanistan and the contiguous Tribal Areas of N.W. Pakistan were the region where this dogma flourished.
So to break your question down you seem to be asking three things did the 1953 coup cause the radicalization of Islam, Did it lead to Iran becoming more conservative and how liberal was Iran anyway.
So to answer the first question no I don't think the 1953 coup can be said to have caused the radicalization of Islam for a couple reason's. First Iran is Shia and a majority of Muslims are Sunni, what is today termed "radical Islam," generally tends to be Sunni groups like Al Qaeda and ISIS are Sunni and follow mix of a radical mix of Salafi and Wahhabi traditions which have an entirely different lineage than anything in Iran. Indeed most groups which fall under the term radical Islam consider the Shia, and thus the majority of Iranians, aspostates. Additionally by the time of the Iranian revolution these ideas were well developed on there own right, and indeed the idea of Wilayat al-Faqih as implemented by Khomeini has little appeal to Sunni groups do to their different relationship with the clergy. The only place they were really able to spread it was Lebanon and even there the Shia are not large enough to create an Iranian style Theocracy.
Now as to the resistance in Mosques I do not know enough to speak about Iran. But I do that in Arab dictatorships this was not to uncommon and in the 1960s and 70s the opposition was much more secular than it is now. But it's not really because they were "to sacred to interfere," most governments in the Middle East even "secular" ones interfere heavily in the management of religion, but it's more that mosques were the only places young men could gather in significant numbers, also while you can shut down all the offices in an opposition party you can't shut down all the mosques, nor can you prohibit people from going to them. This naturally leads the centers of resistance against a regime to be more religious, and it means when new resistance does spring up it comes from the mosques as it is the only place people can actually gather. Whether this is true for Iran or not I can't say. The factors are very similar though with a secular government trying to destroy the opposition in a Muslim country.
Now as to the thing about hijabs,in most Muslim majority countries it's not unusual to see women in big cities without hijabs so it's not so unusual to see the same thing in Iran.
As to the level of conservatism in Iran there are a couple things you should consider.
Iran was never even remotely as conservative places like Saudi Arabia and Afghanistan and still isn't even with it's current government and laws.
The second is to remember a shift in government is not necessarily a shift in attitudes. I may be misreading your question but you seem to be implying that most people in Iran were fairly liberal and now they aren't. It's important to remember that Iran is not a democracy While the revolution brought conservative religious leaders to power I wouldn't say the attitude of places like Tehran have changed that much and if all of the restrictive laws were removed tomorrow it would be likely that we would find them more liberal not less. People cover up because they have to not necessarily because they want to. Look at pictures of woman from Tehran sometime and see how much they toe the line of actually "covering" their head. It's not uncommon to have parties featuring alcohol, as long as it's kept somewhat discrete. So in terms of the liberalism of the population I think it's fair to say they remain quite liberal. At least in the big cities and especially Tehran.