Nintendo would likely start by breaking the series down to the simple elements of what makes a Sonic game. Those being speed and platforming. The story and spectacle would be dialed back as seen in Mario: Eggman is bad, here's his robot henchmen, stop him.

The simplest way to approach this is a 2D side-scrolling platformer. I pretty much already covered it here:
http://venturebeat.com/2014/06/29/why-sonic-should-return-to-his-2d-roots/

You simply cannot have Sonic levels of speed in a 3D platformer and expect it to do well. The faster it goes in 3D the more scripted events you see. That's why Sonic games in 3D get a bad rep for "playing themselves."

Sonic but with gumbas

I think they'd have a good focus on speed and level design, while also having a deeper story than something like Mario.

I would say that Nintendo would probably incorporate the ideas that Sonic Lost World had and simply make them clearer, more focused, and of better quality. Lost World could have been a great game, but it was just really unfocused in terms of level design and had a very odd control scheme. Just heavily refine the ideas in Lost World and give it a more intuitive controls, and you'll have yourself a pretty good Sonic game.

Better.

It would actually be a good game.

Ha ha those parts always reminded my of Sonic games

I think they'd start from scratch; new engines, characters cut and repurposed, and separate 2D and 3D games.

Something like this...

No gimmicks nor character specific levels. Shields would be more prevalent and each would grant unique abilities as well as invincibility for one hit

3D -- Sonic the Hedgehog: insert title (more intricate plotline)
2D -- Sonic the Hedgehog Rebirth: insert title (thin plotline)

Heroes -- Sonic, Tails, Knuckles, and Amy (Amy redesigned with more athletic gear similar to Boom)
Villains -- Eggman, Metal Sonic, Shadow, and badniks

Would be polished.