Word or phrase for showing great skill at something without formal training in it

Solution 1:

Often these sorts of people are called "naturals" -- "Wow, your kid is a real natural at basketball! I can't believe he's never had a coach!"

Such people, especially children, may also be called "prodigies", which mildly-suggests that they were pretty good before training, but doesn't rule out training per se. Some English speakers have adopted the German term, wunderkind.

I think you're correct in shunning 'untutored' for this application.

Solution 2:

a natural-born X TFD idiom

Possessing an ability innately. The term is always used as a modifier before a noun.

As in:

My daughter is a natural-born chef.

Solution 3:

In some contexts "self-taught" might be appropriate. It does not imply that the person was a natural nor does it indicate they were generally untutored.

From Merriam-Webster

self-taught adjective \ ˈself-ˈtȯt \ Definition of self-taught 1 : having knowledge or skills acquired by one's own efforts without formal instruction a self-taught musician 2 : learned by oneself self-taught knowledge