When should I expect the birthday coupon to show up?

I can’t give a reliable number of days from the information here. I’d check the account’s offers around your birthday rather than expect it at a specific time, and make sure the birthday details on the account are correct.

If it doesn’t appear, check again later before assuming you missed it. If the account details look right and there’s still no coupon, asking the game’s support team is the clearest way to find out whether one should have been issued.