Although there's no need to wait for the statement, I wouldn't describe it as "almost immediately". IME, it takes a good few days. The transaction will take a couple of days to reach the credit card part of the account. It then takes a couple of days before it reaches the earned Avios part of the account. It then takes another day or so for that to trip the companion voucher at Amex's end, and then another day or so for that to reach BA. If time might be short, my advice would be not to underestimate how long it could take.