When did the switch happen from straight faces to smiles? Also, why did it occur? Were there some places that took far longer to adapt to the new style of picture than others?
It's not really a question of when people started to smile and more a question of when people stopped having to keep still.
The reason why people didn't smile on early photographs is that the exposures were long. In early photographs the exposure time was as long as 30 minutes. The subjects would have had to keep still and not move for the entire time and as you can imagine no one would want to try to hold a smile for 30 minutes minutes.
As film and flash technology improved this no longer became an issue. I've seen photographs as old as 1880's with smiling subjects, but by the early 20th century camera and film technology had advanced to the point that exposure time was down to a few seconds. At this point it no longer became necessary to hold a pose for longer than a few seconds and you could know smile in your picture.
This has to do more with technology than "style". Early cameras used glass plates which took a long time to expose and the subject had to stay still for about a minute. Smiling for that amount of time is difficult without moving at all and so the subjects tended to keep a straight face. Side fact: Post-mortem photography was pretty popular at the time, especially when a young person had died. This often caused that person to be more in focus than the others as they weren't moving at all. Especially if children died, this was likely to be the only picture the family would have taken.
Around 1900 film was introduced which had much faster exposure rates and Kodak coined the phrases "snapshot" to emphasise how easy and quick photos now where as well as the "Kodak moment" lifestyle to advertise their products. This allowed more people to take more natural looking photos in which people were more likely to be smiling.