Warning: Wikipedia contains spoilers
At the Turkish border Simpson is arrested when firearms are found hidden in the car. The Turkish Secret Police suspect the "tourists" to be revolutionaries planning a coup d'etat (see also History of Turkey) and release Simpson only on condition that he become one of their agents. From now on, Simpson has to spy on Harper, Lipp and the people they meet in Istanbul. Eventually Simpson is hired by Harper, too -- not just to be their driver but to be a member of the gang, too. While all this is happening, Simpson, who is a coward longing for his peaceful life back in Athens, has no idea what exactly they are planning. When the big day finally comes, Simpson, who is afraid of heights, almost has to back out because his job involves climbing the roof of the museum.
It should be noted that Ambler's novel is different from the movie on several counts. For example, in the book there are frequent flashbacks in which Simpson's school days back in England are described, which explains his character and motives more clearly than in the film.
How To Steal A Million (William Wyler; US, 1966) starring Audrey Hepburn and Peter O'Toole is a film with a similar subject-matter.