Assuming the attempt was real and not a staged event for autocratic purposes like suggested above, here's what I'm thinking.
The Tudeh would be still banned, though I am not sure who gets the crown. My guess is that it would be the Shah's younger brother, Ali Reza Pahlavi I who would ascend to the court. He graduated in political science, but I do not know much else about the guy except that he'd be entering as a divorcee though he has a son by this point in Ali Patrick Pahlavi. I don't know much of what could be said except that he would have some preference for the French given the airforce connections and his ex-wife being a French Pole.
Given his education in political science from Harvard, he probably would be a savvier political operator than his brother, but I don't know from there. I don't know if he would okay the repression or whatnot. Perhaps he would work better with people like Mosaddegh and his Harvard connections could be enough to smooth tensions with the US to prevent the coup of Mosaddegh. If the US and Iran remain on warm relations, then the US are gonna disregard the UK's suggestions on the matter.
That being said, if it was a staged attempt that went wrong, I'd imagine Ali would not be happy to hear this though I'm not sure what he would do. If it did leak out the Shah's assassination was basically a staged attmept to justify autocratic measures, this would be an embarassment diplomatically to the crown that the left-wingers would surely use to paint them as the bad guys. Ali Reza might have to jail the archietects of this scheme and anyone else involved on conspiracy and for you know, murdering the Shah. I have no idea if he would unban the Tudeh, but if the revelations came out after they were banned, it would paint them in a sympathetic light. Ali Reza could basically talk to them for negotiations while also expressing what he knows regarding US politics. The main thing would be for Ali Reza to convince Eisenhower that they were not USSR sympathizers. Reza would probably have the Tudeh publically denounce the USSR and reverse the concessions to Russia a few years prior. This along with the aforementioned scandal of the staged assastionation to establish autocracy that went awry could be enough for Eisenhower to disregard British requests and focus on the Thaw.
I don't know if Ali would do the same mistakes as his older brother would, but the original Shah did have a certain personality that probably aided in his downfall. He could navigate through political quagmire of being a leftist member of the First World. I am thinking he would negotiate oil prices with the US (probably not the UK) and would enjoy warm relations with them. It also depends on the Tudeh. In the scenario where the assassination was real, they'd still be banned, but perhaps could reform under a different brand or underground as long as they were smart. In where the assassination was staged, but accidentially succeeded, they'd have to publically shun the USSR to avoid not getting screwed. Granted, the Brezhnev era's return to repression could definitely turn them away for realsies after the liberalization going on in Iran.
This would just leave the conservative hardliners, but given how they were boosted by the numbers of leftists (before backstabbing said leftists), it would be harder for them to do, especially if any of them spewed anti-American rhetoric. Uncle Sam may be a bit annoyed by the lefties in Iran, but they're damn loyal at the least, so the US could provide assistance there against the theocratic agitators.
Crown Prince Ali Patrick Pahlavi could study abroad in America and possibly France as well to gain a more cosmopolitian view and his Polish blood could make him somewhat popular amongst the Poles.
Iran could probably do well as a constitutional monarchy and could breathe easier with the fall of the USSR. Granted, the Afghanistan question would be a biggie. Granted, the Saur Revolution might be averted if the PDPA was influenced by the Tudeh or seeing Iran's success to go more toward gradual development, which could avert alot of trouble in Afghanistan. Granted, it would depend what would happen to the Afghan royal family after the surprise coup. Mohammed Zahir Shah and his family could remain in Iran for the time being or elsewhere. During the Soviet-Afghan War, the Americans and Iranians may instead back Mohammed or his son Ahmad Shah Khan to reestablish order. The Americans probably wouldnt back the Mujadeen as much in the war. Of course, the inter-fighting in the party would happen and after the USSR collaspes, Mohammed or Ahmad could be restored to the throne and Afghanistan could be restored to a secular constitutional monarchy with American and Iranian soldiers.
Ali would probably recognize Israel and not fall want to get involved into the political quagmire there of religious extremism, though they'd probably still advocate for a Palestinian state or something of the like. We probably wouldn't get a War on Terror, especially if Iran pretty much relays religious extremist info to the US to combat it. Hell, they may even pretty much point out how it's backed by Saudi Arabian wahhabism. Additioally, regarding Ba'athist Iraq, they'd likely still get into conflict with them and the Americans would support Iran crushing that.