This is such a tough situation that you're in. Unlike moving a dog or cat overseas, once you involve anything requiring a CITES permit and/or U.S. Fish and Wildlife clearance, things become much more complicated, time-consuming, and costly. (I used to work in a similar field.)
Here are some ideas:
Maybe you can throw a bon voyage party for yourselves, and in lieu of gifts, you could ask for a donation to the Move Harpo Too Fund.
Maybe make it clear to your friends and family that, for Christmas, you don't want a bunch of gifts that you'll end up having to leave behind when you move, or that won't work in the UK anyway, and ask them for money instead.
Or, how about getting really creative. You know how they have "adopt a <insert name of endangered animal here>" charities. Maybe you can do something similar on ebay where people bid to "adopt" your parrot. For the winning bidder, you can send them information about African grey parrots along with monthly updates and photos of him, or if you can teach him your sponsor's name, you can even send them a video clip of him saying thank you to the sponsor.
Just some ideas. . . .
Good luck!