I made both of these discoveries pretty close to each other while running World 1 on the morning of July 11, 2020. I found a 1-1 cycle save, then 57 minutes later, the 1-3 Walkster Warp. At the bottom I'll link to the current World 1 world record, which uses both of these and should be a self-explanatory visual aid.

1-1 cycle save: In 1-1 you can make it to the platform while it's still moving left if you get there quickly, which saves a cycle and removes the waiting for the platform to move back to the right in 1-1. This is pretty easy once you get the hang of it.

1-3 Walkster warp: I found a strange glitch on accident which I call Walkster warp. On 1-3, get to the left of the Walkster, then jump to the platform right above it that the Walkster can hit. Stand on the right side, and the right edge of the flat part of Chompster's body has to be left of the leftmost gray spike. When hit with the Walkster, Chompster will warp straight to 1-4, presumably by hitting the exit warp or the spawn point of 1-4. This is also easy once you get the hang of it. You have to wait a bit for the Walkster to walk back and hit you, but it's overall faster than jumping to the top and moving to the finish. Unfortunately the only other warp I found in the game is getting hit by the Spikejets on the left edge of 1-3, which is obsolted by the Walkster warp.

The 1-1 cycle save lowered my best split on it by 5.2 seconds, and the Walkster warp lowered it by 2.5 seconds.

World 1 world record: