A well done extensive platformer with with variety.
When you have proceeded to worlds beyond world 1, you can go back to world 1, but you can't go to world 2 or 3. You have to make the game remember the progress of the player so you can go to the later worlds again as well.
You did not specify controls properly. The natural position for pause is at P so the placement is good enough, but you did not write it. It is important when you have functions such as retry and level select under there.
You need a mute button. The music is fitting, but listening to it for long periods of time can get bothersome. The sound effects are sparse. A very subtle running sound when holding X with jeans and a floating sound with a kilt, would be nice.
The game gets harder fast. You will get both hate and love for that with other gamers. I like it.
Giving people medals like bronze, silver and gold instead of denying them progress at all if they don't meet your time would probably be wise. People love achievements. If you make some achievements in conjunction with medals people will play your game a lot, even more than they would've otherwise.
An actual main menu is appealing even if it isn't necessary for the game.