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Under Iranian law, all Iranian nationals must travel to and from Iran using an Iranian passport.
Hence, I do not believe they should attempt to leave Iran until they have a new Iranian passport. And once they do
In order to return to the UK youβll need to demonstrate to the Iranian authorities that you have a right to return. At present, the Iranian authorities are accepting the presentation of a British passport on departure if you donβt have a UK visa in your Iranian passport. However, the Iranian authorities could change their position on this and itβs officially illegal in Iran to hold two nationalities.
Therefore, to avoid advertising their British citizenship to Iranian authorities, they may wish to travel via a third country which permits Iranians to travel there with minimal difficulty, such as Oman or Qatar. This way, they won't need to prove they are allowed to enter the UK when departing Iran.
If they do take this route, they should book travel using separate tickets. Potentially it may be safer to book a return flight to the third country and then book their travel to the UK once they've arrived there (and don't use the return leg back to Iran, obviously).