No matter how I look at it, 'Mine and Greg's' seems to be correct. Whenever I have a problem like this, I always break down the sentence into parts.

First, you have the object, which is the 'child.' And then you have the two possessors, 'Greg,' and the pronoun for 'yourself.' Depending on which person you're writing in, the difficulty of this operation varies. First person is awful with compound possessives!

Take the sentence and break it down to just the object in relation to each separate possessor.

"The child is Greg's."

And, since it's first person, and the child belongs to 'you'--

"The child is mine."

Therefore: “I so appreciate you taking mine and Gregg’s child to school today.”

I dub thee correct.

Take it with a grain of salt; my ultra-super-cool qualifications lie within the many treacherous and unedited territories of Fanfiction; I wouldn't be surprised if I was looking at this completely backwards!

Also, when in doubt, try saying the sentence out loud. 'My and Greg's child' vs 'Mine and Greg's child'--which one sounds better? You know which one I'd go with. Just sayin'.