You apply for the upgrade 2 weeks before travel by phone and then you should get an email from the upgrade team telling them what to put in an email to them (inc flight confirmation). My understanding is that you only pay for the upgrade if it is successful. They will take payment by phone.
Personally i would not assume that it will arrive before you fly and apply for the upgrade as in your case you will have been waiting more than 6 weeks. (I am on over 9 weeks since applying with no movement)