Datapoint: I applied on June 11 for GE. I got the approval today (June 23). The technical earliest for the interview for one or two spots was within a day or two, but I needed multiple slots. The earliest I could get was July 6.
I had to wait on three of the apps (due to waiting for the passports to arrive), and I just did them today. Hopefully, I'll have the approval and interview quick enough before we leave.