It would depend on geography frankly. Somewhere like DC and Boston might have recovered a bit quicker since they have a relatively high concentration of Vaults, DC having 7 and Boston having 5. Even if the Vaults opened at different times, them not having an associated experiment would likely give them a similar cultural outlook, which would likely be some bland take on 1950s suburbia. This could make for a good story or a boring one, it would just depend on the setting. Like in DC, I could imagine the Vaults uniting and creating a replacement faction for the Brotherhood and having them get into a conflict with the Enclave instead. I would assume the Enclave would still have a massive superiority complex and be reclusive even if they weren't genocidal which would make for some sort of conflict.
In Boston, it would mean the Minutemen either not existing since the Vaults could just fill the vacuum with more resources or they'd be a real force due to having access to large, well equipped settlements.
New Vegas to me would be the more interesting proposition since a bunch of Vaults around the Strip would be an extra hand of cards at the table. Would they side with Mr. House and form a buffer around the Mojave for him? Would they be a thorn in his side? Would they support the NCR? The Brotherhood? Be Enclave sleeper cells? The story there would be far more interesting.