Iranian oil was nationalized; in 1951 the National Iranian Oil Company took control, and retained it even after Mossadegh’s ouster (even up to today).
> The UK (and US) sought the support of right-wing mullahs, then overthrew the prime minister and replaced him with a dictator.
First, by this point in time Mossadegh had dissolved parliament and was ruling by fiat based on a rigged plebiscite (he claimed that 99.9% of Iranians had voted to give him control of the country[1]). Second, he wasn’t replaced with a dictator. The Shah had been in power since the Soviet Union and the U.K. had forced his father to abdicate over a decade before, and had been in a power struggle with the Majlis for quite some time by that point.
> Eventually even the right-wing clerics and bazaari grew tired of foreign interference and threw the western powers out.
The Shah was (mostly) friendly with the West, but he was hardly a puppet. He was the one who got OPEC to double their prices during the 1973 oil crisis, which hit the West pretty hard. Here's how PBS put it[2]:
> The final blow came in December when the Shah of Iran, ostensibly a U.S. ally, took advantage of American impotence and persuaded the rest of the Organization of Petroleum Exporting Countries (OPEC) to more than double the price of a barrel of oil from $5.11 to $11.65.
[1] https://www.nytimes.com/1953/08/04/archives/mossadegh-gets-9... [2] http://www.pbs.org/wgbh/pages/frontline/tehranbureau/2012/06...