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.
Are you insinuating that the 1979 coup occurred because the Shah was planning on nationalizing Iran's oil business, as P.M. Mosaddegh was doing in the early 50's?