Based on the successful stories here (including ours) most of us got the call very late, like 2 days before the flight. And once they call and take the payment, the passport can be collected later in the afternoon or next day 10am from your local office. But the problem is just it’s not guaranteed that you will get the call.
I saw a post in the Facebook group someone going to the Liverpool office without an appointment as they were flying in 2 days. She said there was long queue outside (people in the same situation) and they were told by the security that they would not be seen. But at the end she was in and got the upgrade done. So I’m not saying it’s what you should do. But in your situation it’s not just a holiday for yourself (I guess there will be guests traveling to your wedding, based on what you wrote), so maybe that worth a try. Even if you can postpone, you don’t know when you will have the passport.