Maybe I shouldn't have been so incredulous. This is from Micheal Heseltine's autobiography:
"In October 1972 Iran Air signed a preliminary agreement to purchase two Concordes for delivery in late 1976 or early 1977 with an option on a third. Six and a half years later the Shah was deposed and for at least two years before that he came under increasing anti-modernisation political pressure."
The Iranian revolution took place in 1979, and Iran Air officially canceled their order in April 1980, the last airline to do so.
The Iran Air office was open today, so I went in and got some better pictures:
Turns out that, due to sanctions imposed by the United States government, Iran Air can only purchase US-made aircraft which are at least 7 years old, and they only can be purchased through a third party rather than directly from Boeing.
Also, the quantity of US-made parts means that it not possible for Iran Air to purchase aircraft directly from Airbus.
Therefore Iran procure a lot of their planes from Russian manufacturer Tupolev, who in the 60's developed their own Concorde clone, the TU-144 - colloquially known as Concordski.
Iran Air's offices are right next door to Aeroflot's:
Indeed I suspect they may even share some 'back-office facilities'.
However, Concordski never sold outside of Russia, mainly due to this crash: