It is a Thanksgiving Cactus, I believe.
The points on the margins are a give-away, as well as the orientation of the flowers (which you mentioned).

Describe the soil, the watering procedure, and the amount of light that the plant receives.

I've had the best luck growing these in a mix that is mostly Orchid Bark. Sometimes I
do add a small amount of Perlite and African Violet soil to slightly increase the moisture retention.

In the long run, it's better to have a fast-draining mix that needs to be watered regularly
than a water-retentive soil that takes weeks to dry out. These plants grow in the crooks of
trees in the jungle, with frequent rains but very little substrate for their roots to come into
extended contact with water.

Don't prune it just yet.

Josh

The soil at the moment is an equal mix of multipurpose compost, sharp sand, and concrete sand (particles range between sand-sized and 2-3mm). There's a bit of bonemeal in there as well.

Generally it got left outside once frost ceased to be a worry (though it's currently indoors on a south-east windowsill and will be staying indoors for at least a month), and it got a lot of light from mid-morning onwards. It formerly received very little water from me unless the soil was bone dry, and it didn't hang onto much because of the shrunken compost (this was the major reason it was repotted). I've yet to water it in its new pot, since it's still damp through from the new mix and the roots got a good dunking when I cleaned the compost off.

Over winter... I can't remember. I think it was kept in an unheated but moderately-lit conservatory / outhouse and not often exposed to light outside natural hours. Maybe watered a little once or twice during that period, but probably not. After reading up a little on the cactus's habits, I'm surprised mine's even alive, let alone flowering.

As Josh wrote, the compost must be open, they don't like a compact growing medium. I grow mine in in a mix of bark, coco fibre, white moss, perlite, vermiculite, some peat and a few other minor components.

When it was kept outdoors, did it receive rain water, or was it sheltered? Schlumbergera do not need to dry out completely before next watering, long periods without water damages the roots. It sounds as if your plant have lived a rather hard life, but if the roots are ok it should be possible to save.

It was left open to the elements, and since the thicker roots were supple and hard the plant itself appears to be in fairly good condition considering its treatment. The soil's probably overly sandy, but the plant can stay there for a while until it settles. The 'pot' is transparent, so I can keep an eye on the soil directly.

The mix is all wrong for this plant. While it may be a cactus, it's a jungle cactus, not dry arid land cactus. Sand & peat are particularly wrong for this plant.

In nature it grows in the crotch of tree branches, getting very litle mix, mostly leaf debris & whatever animal poop, etc. falls onto it. The plant is exposed to frequent & fast rains which come quickly leave quickly & the plant mostly dries out quickly.

Looks, sounds like your poor plant has been generally neglected & dying of thirst. This isn't a plant one cuts back like a Jade (besides you don't have enough material there to prune), forget that.

Pls. start watering it more, give it one big watering & hang it over the sink to drain & then I'd switch the mix completely. Try an AV mix w/ extra perlite, maybe a little bark.

In a month or so you could try some fertilizer if the plant has shown new growth. But pls. take it out of that awful, all wrong for it mix (which stresses the plant even more). Pls. make sure this container has a drainage hole.

These plants can be tough & resilient, w/ some correct & consistent care, it can come back pretty nicely. Pls. read up on their care (including the FAQs); it doesn't matter of Xmas or Thanksgiving cactus to improve its care & feeding enough to really help it come back. Good luck.

I haven't been able to find perlite anywhere round here. Can I use rounded gravel instead?

No, gravel is too heavy, but I guess better than nothing. I see you're in the UK, don't you have nurseries that would sell this, especially in the Springtime?

Heavy in weight, or heavy in composition? If it's the first it doesn't matter.

There is a nursery round here (it's part of the Wyevale group, if that information helps any), but I think it's the one place that slipped my mind when I was hunting for ingredients (I was repotting something else as well... that'll probably have to come up as well, but it's a big desert cactus and fully green and neat. It can wait its turn). I might have time tomorrow to get there after work.

Is there anything else essential I should pick up with the perlite if it's there?

It really depends on what plants you wish to grow, other than the holiday cactus. There's loads on info. here on mix (soils), amendments of various kinds.

Then again, if one is new at this, it helps to keep things fairly simple & focus on observing the plants & responding to their changes.

Finally got to the garden centre, and while they didn't have any African Violet mix I had enough ingredients to make something.

2pt perlite
2pt peat moss
1pt vermiculite
1pt grit / gravel
All of those were screened (the ruler in the photo is a little indistinct, but the intervals are 1/16"). I'm just going to go ahead with the repotting -- the mix might not be perfect, but it sure looks a lot nicer than what I've got.

Here is a link that might be useful: {{gwi:548242}}

{{gwi:548244}}

{{gwi:548246}}

That shouldn't be too bad, actually...
When you wrote "peat moss," my stomach immediately dropped.
But then I looked at your pic, and it seems the moss is the
long strand form.

The African Violet mix is not essential; I've used bonsai potting soil, as well.
The potting soil (regardless of type) is used to "clog" (or bridge) the spaces
between the larger particles of bark, perlite, and grit, thus increasing moisture
retention. You can adjust the organic component/s of your mix to suit your climate...
finding a happy medium, if you'll pardon the pun! ;-)

Now that you've re-potted, do not place the plant in direct sun. Let it recuperate
in a shady location. If it wilts or droops, wait until it perks up to return it to a brighter,
sunnier location. I always keep my
Holiday Cactus in a partially-shaded/filtered area. They don't need full sun, and will often
yellow or blanch or burn from too much sun. The right amount of sun, however, will induce
a wonderful mix of green, yellow, and red color.

The recovery should be swift.

For arid cacti, I would make a mix of 1 part bark, 1 part perlite, and 1 part grit.

Josh

It's strand form, yes. I thought 'peat moss' always referred to the moss itself, or is it sometimes used for decayed sphagnum?

The Schlumbergeras are now sulking in the washing room, which is unheated and lit through a hedge in the afternoons (I've put up a board to cut the sunlight out). There turned out to be three cuttings in the pot, one of which ended up in a section of egg carton because I didn't have any suitable 'proper' receptacles. No drainage worries there, at least. : )

This is straying off-topic a little, but I have some play bark available. Given that I have a very large cactus cluster to repot (the specifics of which I'll leave for a fresh topic), will the large chunks be acceptable?