If there was an easy way of getting to Edinburgh from Spain, I would suggest it.
Open up Google maps because it's time for a geography lesson
The easiest ways to get an animal into the UK is on the shuttle (which as I've said, requires a car or other motor vehicle) or ferry. These go from Calais in France and will get you to Folkestone (shuttle), Dover, Southampton, or maybe Portsmouth (ferry). Because the original poster is in Spain, they're going to have to fly, get a train or drive to Calais.
The UK's main internal airline, FlyBe, has a blanket ban on animals. Animals cannot travel as cargo or in the cabin. This means that after entering the UK, the original poster will have to either drive or take the train to Edinburgh. If they are taking the train, they first need to get to London. Because trains arrive and depart from different stations scattered around the city, they'll either have to navigate London's underground with two carriers plus their luggage or get a taxi. London's underground is one of the oldest underground railways in the world and while most stations have escalators, fewer have lifts and true step-free access. They could get a taxi across London but central London traffic is horrific and it will be expensive - a minimum of £40 depending on traffic and time of day.
From London, it's a five hour train ride to Edinburgh. My favoured way to travel is the sleeper train. Pets are allowed in the cabins - however, the train departs at about midnight and gets in at about 6am so is going to seriously disrupt the birds.
All in all, this is going to be well over a day of travel, navigating unfamiliar transport, quite a lot of switching between modes of transportation and very stressful for both human and birds. It's not as simple as hopping on a direct train.