The title really says it all, is it legal to swim in the lakes in the Austrian Alps?
It is allowed if there's no sign declaring it as
Naturschutzgebiet/Naturschutzzone this is german for
natural reserve if there's no such sign or you know for sure your not in such a reserve, it's safe to swim legally. However, best practice is to ask at the local tourist information center.
This is a copy paste from wikipedia:
The right to roam in Austria, particularly in forests and mountainous areas, is called Wegefreiheit. Since 1975 the right to roam in forests is guaranteed by Federal law. In particular, walking, running, hiking, and resting are automatically allowed to the public in most forest areas. However, horse riding, bike riding, and camping are not, and may only be practised with the land owner's permission. A large proportion of the forest area in Austria is owned by government bodies such as the Österreichische Bundesforste, but the same restrictions still apply. In some circumstances forests may be closed to the public for environmental reasons. The situation in mountainous areas is less clear, and differs from state to state. Some states, such as Carinthia, Styria, and Salzburg guarantee a right to roam in mountainous areas (usually defined as above the tree line), for all recreational activities. In other states, such as Tyrol, Lower Austria, and Burgenland, no explicit right to roam exists and land owners reserve the right to deny access. In practice, however, such restrictions are rarely enforced, since mountain tourism is an important industry in Austria
Sources : Wikipedia (article Freedom to roam)