So after a few days of testing, I finally found a consistent way to pull of the skips glitch for both Sonic and Tails. I'm surprised that I didn't find this earlier considering how obvious this should have been to me. Open up your game, and do a run as Sonic. After this run, choose to play as Tails or to play as Sonic, depending on which character you would like to use the glitch with. It's really that simple. The reason I couldn't get it to work consistently before is because I was doing the set-up as Tails, and the characters would constantly fall under the ground when re-entering the game.

How much does the skips glitch save?

Although I don't know the exact timings for Sonic, I know that the glitch skips 4 zones, 2 cutscenes, 2 power-ups and 1 boss fight. I reckon this glitch could save around 10 minutes for Sonic.

The glitch isn't as significant for Tails, as it only skips 1 cutscene and 1 boss. However, the boss is incredibly inconsistent, as the number of hits you get in 1 spring bounce is completely random, so it skips an incredibly inconsistent boss fight.