I can't really see a defined curl until I have at least 2 inches of new growth. Otherwise it just be fuzzy. I could proabaly see it sooner but I've never cut my hair that short to know how it would be with no relaxed ends.
It may take a little longer for the curl pattern to come in. Also products make a difference too. Some leave the hair puffy and undefined while others define the curl.
For my kids with the very tight curls/coils, the pattern is recognizable at about 2 inches. For my kid with the looser curls, the pattern was clearer around 4-6 inches. I have a mix of patterns, and the tighter patterns are also visible at shorter lengths than the looser ones.