While the month is never explicitly stated, there's mention of August traffic jams.

“Matto, caldo, soldi, morto… girotondo…”

A continuity error later on in the movie makes it August 4.

“I'd become a sort of a reverse zombie. I was living in a world already dead, and I alone knowing it.”

– John Triton

“You'll go under like all the others.”

– Judith Denning

“I don't talk to guys wearing aprons.”

– Earl Macklin

“It's a lot of work, ya know, just staying alive.”

– Rosemary Scott

“Twelve boxes of cornflakes – all for himself?”

– Jack

“On the night of July 25, 1958 I killed Ladislaw Zukarevitch, antique dealer, my mentor, my master in the art of polishing diamonds, my tutor at Warsaw Theological School.”

– the student