Generally speaking, it's no pay, no fly. When Britain doubled their departure tax a few years ago, it explicitly applied to tickets that had already been booked -- so when we showed up to fly home on tickets booked before the increase had been announced, we had to pay $120 in additional taxes for the three of us. Fortunately we knew about it and just paid it at the check-in desk.

But the other aspect of Britain's departure tax is that it's a £80 ($160) tax that happens to have a 50% rebate for passengers in the lowest class of service offered on the flight. If you suddenly upgrade yourself you lose the rebate, so you're going to have to pay the other half of the tax in addition to whatever other fees you're paying to upgrade.