They're so lovely aren't they!
I've got a couple of Christmas ones, but they don't flower reliably, I've no idea why, though. Is it more watering or feeding they require?
I will tell you the care regime mine get, which mimics what my mother used to do. She had a very large Christmas cactus, and she always got it to bloom reliably. I say I try to mimic what she did, as I never got precise instructions from her. I seem to be close enough though.
My mother's Zygocactus grew in a 10-inch diameter clay pot, the plant was at least 2 feet across. Who-knows-what kind of soil. I never saw her repot it. The plant in my photo is in a 4-inch clay pot, and has been in that pot for years. It is likely very root-bound. I don't repot, but occasionally top off the soil a bit.
In spring and summer, mine goes outside. To be on the safe side, put them out after your last frost date, leave them out until autumn, pull them indoors before the first frost. The real truth is that mine are sometimes put outdoors a little too soon or left out a little too late, and a light frost does not seem to hurt them, but I don't recommend it.
When outdoors, water and fertilize like other houseplants you have outside. In the UK you probably get enough rain that watering may not be needed if the plants are exposed to the rain. Give it soluble houseplant fertilizer once a month. In reality, I usually am casual about the exact NPK ratio of the fertilizer, and how often it gets fertilizer, as long as it is growing well. Mine in Georgia are put in full sun.
Autumn. What my mother did: when it is too cold for them outdoors, bring them in, put in a room away from direct sun, and cool to cold. My childhood home had very poor heat in the bedroom areas, dim lighting, she used that space and witheld water completely. I think she did that for maybe a month or two. What I do: our October gradually cools and becomes dry, I leave my Zygocactus out until either it gets too cold or until buds form.
After the cold/dry/dark treatment my mother would put the plant in a south-facing window, and take her 1-quart watering can, add a half teaspoon of plain household ammonia to that, water with that solution, and water weekly until blooming. Mom grew up during the depression, and her family didn't spend money on houseplant fertilizer, but the ammonia provides nitrogen. I just water, with or without houseplant fertilizer, when the plant comes in, and water weekly.
Thats it!