Restaurants in Europe serving Turkish food have of course been around for as long as there have been Turks living and working in Europe. However, the widespread popularization of Kebab as a staple of European fast food is a more recent phenomenon, traceable to the 1960s or early 1970s with the introduction of the Doner Kebab (the Kebab pita sandwich). Many believe that the Doner was invented and popularized in Europe by a Turkish immigrant living in Berlin named Kadir Nurman, who set up a famous street stand selling the sandwiches in 1972, while others have credited Mahmut Aygun of doing the same in 1971.
Whether or not either is the true inventor of the Doner is controversial, but most seem to agree that the Doner was introduced in Europe at around that time and Kebab restaurants quickly became ubiquitous throughout Europe as Doners gained in popularity.
As a followup, there seems to be quite a difference between the döner kebab in Germany and the "kebab and curry" in the UK. Is the origin story for kebab in the UK markedly different?