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TLDR: The Great British Baking Show microwave is the Swan Nordic Digital Microwave (Amazon UK).
Swan 20L Nordic Digital Microwave - 4 colors - 2 awesome ones.
(Amazon UK)
The Great British Baking Show microwave is the Swan Nordic Digital Microwave (Amazon UK). These simple little microwaves from Swan come in 4 colors and have a "Nordic" look that's unique to the Swan Nordic line.
The Swan Nordic Digital Microwave (Amazon UK) is a nice little (emphasis on little) counter-top microwave. Its has a 20 Liter capacity and the cooking platter is 270mm (just over 10.5 inches), and tops out at 800W. Lets compare that to a normal installed microwave - they're generally have more like a 50 Liter capacity, and a 12.5+ inch platter.
If it means anything to you - the Swan Nordic Digital Microwave (Amazon UK) has a MAFF E class rating- which basically means it cooks things faster than things with other letter ratings. (The letter ratings are used in the UK and confuse everyone there too, as they mean something different for all other appliances.)
The microwave has 6 different heating levels, a 60 minute timer, and... a cooking end signal. Woo? It feels like they started reaching for marketing material. It feels like more honest marketing materials would have read "comes in 4 colors, looks pretty, has a cool fake wood handle and dial, and it does not take up much counter space." But I guess this is why I'm not in advertising.
The Swan company was formed by Bulpitt & Sons Ltd in the 1920s in Britain. They started making electrical appliances in the 1930s and actually made the first immersion element for a tea kettle and the first electric toaster. After this Swan was purchased by various other entities and eventually returned to ownership by the Swan employees in 2008.
Their stated mission is to "make dream on-trend kitchens a reality for our customers, by offering affordable iconic cookware and appliances collections that boast stylish design and outstanding performance. We are a lifestyle brand for everyday moments."
Here in lies why they likely landed in the tent. Their Nordic line, introduced in 2017 looks pretty nice in my opinion. It has simple lines, and comes in 4 colors. Their Pine Green and Blue colors both have a heavy earthy feel, are fairly unobtrusive, and maintain uniqueness to Swan. The colors fit into the tent seamlessly and almost go un-noticed, flowing nicely with the bold colors on the fridges and stand mixers.
20L is less than half the size of a built in microwave. Those are 10 inch tortillas.
(Amazon UK)
The Galanz retro microwave has a splash of color, but is not nearly as chique.
(Amazon UK)
This is another instance where if you're in the UK you're in luck! Enjoy that healthcare... and access to the Swan microwave! The Swan Nordic Digital Microwave is available on their website and Amazon UK.
If you're in the US or anywhere outside of the UK you're kind of out of luck. No major retailers sell the microwave, and Amazon does not carry it. Your best bet may be to find one on Ebay. However, you're probably going to find it from a UK retailer and pay some serious shipping (think ~$60). You may come across the microwave for roughly 50% of what you'd expect to pay on some sketchy websites, but I'd really suggest not risking that endeavor.
Normally I have some awesome options that I fully support. But if you're looking at the Swan Nordic Digital Microwave (Amazon UK) you probably want it for its looks. And honestly, no one else really has this sharp and clean of a look. Galanz, the same people that make The Great British Baking Show's refrigerator also make a retro microwave with a splash of color, but it kinda looks frumpy in my opinion. So your best bet if you're not in the UK is going to be Ebay.