At the time of writing the history of Islam and the Muslim figures in the 19th century, did the European institutions and Orientalists go back to copy the same sources and writings known that time either in France or in other European countries?
19th Century Orientalists relied on works like Abu Ja'far Ibn Jarir at-Tabari's "Tarikh Al-Islam", which was translated under the supervision of Dutch orientalist, Micheal Jan de Goeje. This particular work is a favorite of orientalists, spawning many efforts in translation; probably the most notable of which is that of Franz Rosenthal. Other Arabic (primary) sources include Ibn Al-Athir's Al-Kaamil.
You would be able to identify the sources through key works of notable orientalists in that time period. Probably one of the most influential Orientalists of the time was the German, Carl Brockelmann, whose work on "the History of the Islamic People" has been translated in the early 20th century. Likewise, one can find the references used by the Hungarian Orientalist, Ignac Goldziher and the Austrian, Alfred von Kremer.
In the 19th century, a group of orientalists appeared on the history of Islam and its themes were: Van Floten, Van Gelder, Theodore Noldeke, Karl Brueckelmann, Flahausen, Mayer and their works on the life of the Prophet Muhammad, the Holy Quran and the history of Muslims. In the 20th century, the most famous of them was Montgomery Wat.