I agree with a lot of the improvements suggested here, but there's one thing I would like to add. Characters that are likely to appear next game, like Trowzer or Dr. Puzz, should get expanded roles, and their relationships with Yooka, Laylee, and other characters should be more developed.
However, I'd rather the relationships be developed not so much through character dialogue, although good writing is always a plus. Instead, their relationships would actually be developed by the gameplay itself.
For example, Trowzer only gives the main characters new moves in exchange for quills. In the next game, what if Trowzer started offering useful services as well? For a fee, Trowzer could revive Yooka and Laylee, then return them to the last safe platform they have stood on, in the case that they fall into a bottomless pit. For humor's sake, he could hop into the scene with a nurse's cap and syringe, and give them a shot to wake them up. An upgraded version of the service could give him a doctor's coat and a defibrillator, and Trowzer would shock them awake for temporary additional butterflies on the health meter. This would probably require a reworking of quills, or whatever currency that Playtonic uses in the next game. It would definitely give a good reason to collect quills though, aside from earning a Pagie. On top of that, it would show that Trowzer has actually made progress on expanding his business, and the costumes he wears for different shops and services would add some charm to his character.
As for Dr. Puzz, she could do research on creating new contraptions that give Yooka and Laylee a fighting chance against the antagonists of the next game. They could replace some of the more excruciating mini-games, like the golf courses (flashbacks to The Black Hole intensify) and they would show that Dr. Puzz is more invested in the effort against V.I.L.E than ever. She could also be at odds with Trowzer, should the snake salesman start selling some not-so-legally obtained technology in his own shops.
Haven't thought of stuff for Rextro and Kartos, but they should have the same sort of improvements as well.