Though Bollywood is a term coined in the 70s, the question here means the first movie to be entirely shot abroad. There is little evidence of movies shot abroad (especially prior independence), still India in Africa a.k.a. Africa Mein Hind can be called as the first Indian film to be entirely shot abroad (Africa). The movie was released in 1939.
The film was directed by Hirendra Kumar Basu (Hiren Bose) and starred B. Nandrekar, Urmila Gupta and Vidya Devi. It was an action film.
After Africa Mein Hind, Ashok Kumar starrer, Naaz had a few scenes which were shot in Cairo, London and Egypt.
Ten years later in 1964, Raj Kapoor shot a few scenes and songs in Paris and Switzerland. The film was Sangam. The film starred Raj Kapoor, Rajendra Kumar and Vyjantimala in lead roles.
Image(s) Courtesy: Google