One thing I do to speed up rolls in general is to not reveal the CR. I figure out the DC while they are rolling and tallying up the result, so that the game doesn’t stop while I think about it. If they roll particularly well, I just give it to them.
For combat, I’ve been doing something I picked up from reading Justin Alexander’s GMing advice. Instead of rolling initiative before combat, he suggests rolling at the start of the session and at the end of combat (for the next one). During those times, people are usually focused on bookkeeping and/or decompressing, so the rolling and recording initiative doesn’t slow the game down as much. More importantly, you’re no longer signposting combat. The transition happens much more naturally as an extension of RP, the recorded initiative informing the order you ask players what do they do. It also seems to work very nicely with using Success with a Twist fully in combat.
Last session, my PCs got into a fight with an octo-squid. That’s a large creature in my setting that’s sort of a cross between the two. It had managed to make a home in an inland lake, and people were having a problem with that. The PCs started out carefully scouting out around the lake trying to get a better view. When it saw them, we just kept playing it out naturally until one of them managed to get a little too close. With their initiatives in mind, I continued going around the able asking them what do they do next. Sometimes they succeeded, sometimes they failed and I got to do something cool. One of the characters is a bard who can influence people with his voice. He tried to call the creature out of the water, so it would be disadvantaged on the land. He rolled terribly on his roll, so I decided he’d get what he wanted, but the danger would snowball. The creature’s tentacles shot out of the water, grasped some trees, and hefted itself up on top of the party. Apparently (and surprisingly to the PCs) comfortable in the trees, it grabbed the party’s brute and chucked him back into the lake. The PCs managed to beat it soundly, but everyone remained engaged the entire time. I credit much of that to keeping our regular cadence even when a fight broke out.