Britain had been involved in the oil business in Iran since 1901, when William Knox D'Arcy, a millionaire London socialite negotiated a 60 year oil concession with Mozaffar al-Din Shah Qajar who was the Shah of Iran from 1896 to 1907. Britain held the majority of the shares of the oil business in Iran, in which Iran received very little money in return. The fact that Britain got America to do their dirty work for them in 1953, and through a coup, forced out a democratically elected Prime Minister just to keep control of the oil business, certainly played a role in what has gone on in Iran since then.
Britain and Russia invaded Iran during WWII to protect Britain's oil control, because they feared the Shah, who had already proclaimed Iran's neutrality, was getting too friendly with the Germans. The Brits didn't give a crap about the country. They just didn't want Hitler getting his hands on their oil business. If the morons leading Britain, and France had honored the agreement they had made with Russia to protect Czechoslovakia, and taken out Hitler and his underwhelming army at the time, instead of going to Munich, without even inviting Russia to the meeting, and handing Hitler that country, there may have never been a WWII.
Yes, the UK was involved in Iran. As was Russia, both as imperial Russia and the USSR. Yes, nationalization of the oil industry was popular in Iran due to the one-sided contract that the former regime had signed. But YOU failed to mention that the Shah was going to nationalize the oil industry in 1979 when the Consortium Agreement of 1954, that he had negotiated which had actually increased Iran’s share of the profits, was set to expire.
In fact it was Mosaddegh who was attempting a coup. He refused to obey the Shah’s legal order to leave the office of prime minister, illegally attempted to dissolve the legislature, conducted a rigged referendum, and finally lost the support of just about everyone.