I earned one in 2021, reaching the required minimum spend in August 2021. the BA website had the same language that you are quoting now, about 4-6 weeks after the credit card statement date.

However, I believe the companion voucher appeared in my account shortly after the credit card statement date, which was early September. Not right away, but within 10 days or so. I did not get any notification by email; I just happened to be reviewing my BAEC account and there it was.

Hopefully you will see yours soon!