The official "world-ending" time in Fallout is 9:47 AM, October 27th, 2077, on a Saturday. Though obviously that time shifts depending on geographic location and order of bombing targets.From the radio announcer in the trailer and the panic in people throwing everything into their cars and running down the street, we can infer that nukes have started striking other locations, and the population knows Armageddon is racing their way.
Since China is doing the nuking, it is likely that targets in the western part of the continent, like Alaska and the West Coast are among the first to be hit. Depending on when the first warning was received that missiles were incoming - either if they were detected at launch, or if the first warning was the initial target impacts - then the people in Boston had 15-30 minutes of warning that nukes were coming. This assumes that ICBMs in Fallout's universe travel the same speed as ours in the real world. (ICBMs average 3 miles a second. Slightly slower at first, and slightly faster before impact.)
I would assume, that knowing this, the government and Vault-Tec would already have the Vaults stocked, as it would be impossible to get supplies for 1000 people moved in in less than half an hour.
One interesting thought I had: From the leaked documents, it appears our protagonist is an important figure and is somehow involved in the military, and is giving briefings. What if our protagonist, the player, is the Overseer for Vault 111? Nothing really to back that up, but I think it would be a great change-up from the usual "You come from a Vault" opening.