I'm not convinced that they actually did. We know that they planned to, but all pre-existing lore suggests that the Great War was started by China, with the US detecting 3 unidentified submersible objects off the coast of California at midnight on October 23, 2077, and then a squadron of Chinese planes crossing the Bering Strait into Alaska at 3 am. Then at 9:13 am the US detected 4 Nuclear launches with their automated system, confirmed to be from China by NORAD at 9:17 am, those bombs then landing in the US a half an hour later at 9:42 am on the east coast, and shortly afterward on the west coast, with the entire US bombarded with nukes over the next 2 hours. So unless Vault-Tec somehow got ahold of a Chinese plane squadron and 3 Chinese subs, it seems unlikely they actually started the Great War.

Not to mention that Barb Howard's whole plan was making sure that her family got into a good vault, and we saw that on the day of the attack, she had allowed Janey and Cooper to go a birthday party, as a high ranking Vault-tec exec she surely would have known that the nukes were dropping that day if Vault-Tec was behind the attack, and never would have allowed Janey and Cooper to attend a birthday party when the bombs were supposed to land. No, I think the war started on it's own, without direct action by Vault-tec. 

I also agree that the show doesn't make it clear cut that Vault-Tec dropped the bombs, all Vault Tech said is they could drop the bombs, I believe that scene was more for your average viewer new to Fallout to show to them how sick Vault-Tec are but I believe it's likely still that someone beat Vault-Tec to it. Us game folk know how sick Vault-Tec are but that scene serves to show both Cooper and the viewer quickly.

Number one reason is like you said, Barb is clearly very high in Vault-Tec and yet the bombs dropped when Cooper had her daughter, the only way that makes sense if if she was acting this entire time and didn't care about her own daughter or someone else within the company dropped the bombs which seems unlikely as Barb looks like a leader in the company.

Secondly, to my knowledge, there are unfinished vaults, suggesting that Vault-Tec was unprepared and not yet finished, so why would they drop the bombs before all the vaults were ready? I think Vault-Tec was going to drop the bombs but things escalated more quickly than they expected and then the bombs dropped, I think it's still vague who dropped the bombs.

As for Shady Sands, Todd Howard has stated that "the fall" is separate from "the nuke," which happened after New Vegas.

Really looking forward to Goosey's first encounter with a Death Claw.

That's what I suspected but holy shit the New Vegas fanboys losing their damn minds at a single date on a whiteboard.

How I interpreted it is a "Fall" of something isn't necessarily only reference to the final date in which something was completely destroyed but the series of events which lead up to that collapse, in that sense, the "Fall" of Shady Sands in this context would be when things started going bad for Shady Sands which led to them eventually being nuked.

That was my interpretation but there's multiple other explanations that work.

I want Thaddeus to encounter a Death Claw and somehow win, Lol.