Anyone who has used the Unchained Revised Action Economy will be able to tell you more about it.
I personally don't think it will speed up play as much as making the system more intuitive, as well allowing for more interesting turns (particularly for martial characters).
The action economy itself is not what splows the game, but rather the amount of dice rolling and math that is done. So I think the promise of faster, more flowing combat will be adressed in streamlining the math and perhaps making individual abilities more intuitive to use.
What always slowed the game down for me was having to go back an look up a rule whenever someone tried something non-standard or forgot how their ability worked (or of we didn't know how it worked in this particular scenario).
How the rounds were stuctured was never really a factor. So I don't think that the new three-action system will change the speed of play.
The larger factor will be if they make abilities more intuitive or if they make it easier to try non-standard things. 5e handled it by emphasizing Rulings (don't know the rule or it doesn't exist, hey GM, it's your job to make it up so game play isn't slowed down).
I'm curious to see how PF2 handles it. But from what I've heard of the live play podcast, it does seem like there's going to be more generic skills and actions to help speed up play while still allowing for the complexity of rules that Pathfinder is known for. I'm hopeful.