*Pet* birds from many countries are still allowed into the UK at this time. This can change at anytime. The paperwork can take months, so I suggest you get started asap!
We have an African Grey, a Green Cheeked conure, and a lovebird and I wouln't move without them. so I know how you feel.
We moved from the UK to Singapore last year with our 4 parrots. In April, we'll be moving back to England with our 4 parrots. There is a lot of paperwork to be done and the EU has now set out some new regulations regarding the import of pet birds.
You'll need import and export licenses from the 2 countries involved.
You'll need CITES export/import permits. Start looking for receipts, hatch certificate, microchip info, legband info now because you will need this info to prove that your Grey is not a wild caught bird. It's possible that your Grey may need to be microchipped, if not done already. Our Grey was already chipped, so it wasn't an issue for us, but I'm sure one of my friends had to get her cockatoo microchipped for CITES. (The more commmon, less expensive birds probably do not require a microchip--our lovebird and conure did not.)
You'll have to fulfill one of the 4 (new) conditions set out by the EU. You can obtain this informaiton from DEFRA.
You'll also need a health certificate for your grey before travel (the health check usually needs to be done no earlier than 48 hours before departure) and they will have to be checked by a DEFRA approved avian vet at least twice during the 35 day quarantine period.
Currently, DEFRA are still allowing pet birds to be quarantined in your home (as opposed to an official quarantine facility.)
Most of the paperwork needs to be done in a specific order (meaning in most cases, you have to be granted one permit before applying for the next). Most of the permits or licenses have 30-90 validity periods, so the timing of each needs to be right.
We are using a pet relocator to handle all of the paperwork and details, which I highly suggest you do, if it's within your budget. Otherwise, DEFRA is quite helpful.
Also, some airlines are not allowing birds on the planes, so you may have to do some research on that aspect. We'll be using BA from Singapore to the UK--not the best airline, but they do allow birds at this time.
If you want to correspond offlist, let me know.
Good luck!