It's been so long since I joined I honestly cannot recall. Someone else may have more recent experience but I'd be surprised if it was more than a week or two.
My Qantas Club card arrives about a week after I've passed on the paperwork to my employer so I can't imagine it being a long turnaround.
Give it a couple of weeks after you've sent the application and you could always call the FF service number if you need too.
IIRC, you receive a welcome email upon joining that shows you're officially registered and allows you to print a temporary card to use in the meantime.
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