Mosaddeq sought fairer royalties for oil from what is now BP but what was then the AIOC after decades of tension and a decrease in Iran's royalties (with increasing British revenues) in the 1940s, ultimately culminating in the nationalization of AIOC in 1951 [1].
Relations deteriorated. Britain isolated Iran through sanctions and oil embargoes. The US sided with Britain but initially rebuffed attempts at a coup, I believe initially under Truman but Eisenhower was also initially reluctant.
Britain did argue that nationalization of oil and other British interests in Iran was Soviet-led and made an argument to Eisenhower's SEcretary of State that a coup was in the interests of fighting communism, something the administration was likely more receptive to given the Truman doctrine and "containment". The Korean War was ongoing at the time.
So did Britain argue this was to fight commmunism? Yes. Was it really? No. It was about Britain's oil interests and colonial ambition. It was no more about fighting communism than invading Iraq in 2003 was about spreading democracy.
Fears of the USSR played a much bigger role in the 1979 Revolution where the US got their then ally, Saddam Hussein, to release the Ayatollah Khomenei from prison to try and make Iran fundamentalist rather than falling into the Soviet sphere of influence.
As for any election abnormalities, nobody cares about that. Like, at all. It's undeniable that Mossadeq was immensely popular in the early 1950s for his stance that Iranian oil should benefit Iranians, first and foremost, rather than a colonial power.
[1]: https://www.ebsco.com/research-starters/history/iran-nationa...
