How long depends on how busy the
GNAS office is. It can also depend how quickly the tournament organiser returns the paperwork to the
GNAS office after the tournament. There is a time limit for this (30 days I think), so it shouldn't be a big factor.
I've usually found it can be any thing from 1 to 3 months. I received my 1200 FITA star within 6 weeks this year, but know of others how have waited longer for awards this year.
They are issued during the season, but obviously there will be busy periods during the season as the number of tournaments ramps up - so delays will be longer.
I guess it's also possible that some delay may occur if
GNAS office does not have the badges in stock - given that some will be in greater demand than others.
If it does drag on beyond 3 months, it may be worthwhile ringing the
GNAS office to check the claim was received and then following up with the organiser if necessary.
Also, the claim form requires the archer's signature but is poorly designed, so this often missed - invalidating the claim. I know I've had to chase after archers to sign their claims before they leave the ground after a tournament.