Yeah, it's going to depend on what you intend to write down.
If you're just writing lyrics with chord names over them, then it doesn't matter what type/tuning of ukulele plays it. For that matter, it doesn't matter what instrument plays it (guitar, uke, piano... whatever). As Lori mentioned, a G chord is a G chord regardless of tuning or instrument. What will change is how that G chord is played. But it still gets the letter G as its name.
If you're writing standard notation--a staff with notes on it--then it's the same deal. It doesn't matter (generally) what instrument plays it. Notes are notes. Standard notation just tells you what to play, not how to play it, so the type and tuning of the instrument used is not so important.
Pretty much the only time you'll need to worry about tuning is if you're writing down something that tells you how to play a chord or note. So if you're including chord diagrams along with the letter name of the chord, then you will need to take tuning into account. And if you're writing down tablature, then you also need to take tuning into account. That's because chord diagrams and tablature tell you how to play things--where to put a finger on what string and fret. That definitely varies with tuning.
JJ