When a nuclear war erupts on the first day of summer, a astronaut must guide a group of survivors towards safety, for the survival of humanity and his return voyage home.

The goal is definitely unclear, as is when and where the story is set.
Is it during the nuclear war?
Do they want to survive? To save humanity? To return home? (Where are they if they're not "home"?) It's OK if the script has several of these things tied in together, but I think it needs to be streamlined and clarified for your logline. What's the most important of these goals (that will facilitate the others)?
Are they in space or on Earth? (If they're in space, are the survivors his crew? If they're on Earth - as Harvey suggests - are they are rag-tag bunch?)

Couldn't have said it better, Harvey.

I was about to say something like 'where's the antagonist?' but I believe that by describing the survivors as 'rag tag' can imply that the struggle could be between the members of the group, either against one another or against their leading astronaut. Like in the movie 'Cube' (one of my favourites), when the baddy turns out to be one of the gang, but not until quite a while in.

Unclear as why it matters that it is the first day of summer and where the survivors are. Are they in space? Assuming so because of the vocation but unsure. Also, what problems will they encounter en route?