Varies wildly between children.

DD - 00000: two weeks
0000: about 3 months
000: same
... Then I don't remember but I know she was in a few 1s by her first birthday but they were baggy. She's long and lanky

DS - 0000: 2 weeks
000 - about 6 weeks
He is now 4 months and is wearing 0 / 00 ... I find there's quite a lot of variation between brands so he's even in a couple of 1s. He's my 95% percentile Chunky Monkey though, so he's in these bigger sizes very early!

I think it's probably best to just wait and see how big bub is and how quickly they grow. Had I planned ahead I would have bought more stuff for the cooler months that he never would've used and would never have dreamed that he'd need size 0 t-shirts.