Calorie deficit is not working

Go to the doctor and see what's happening. In theory, and practice, a caloric deficit is literally all you need.

This doesn’t make sense, if you are eating so little you must be burning it off.

You must be unhealthily thin. Your body cant day after day live off so little energy, your body naturally consumes more just to maintain its own structure.

If you are not exercising even a little bit, then I'd expect your body to get lazy in how it processes intake. If your routine isn't demanding much short term energy use it makes sense your body would feel more confident stashing it away long term.

How are you working out a BMR of 500 btw? for comparison mine is 3000,

I think as others have said your low calorie diet is causing the fatigue and bad skin. When you don’t eat enough your body goes into starvation mode and you don’t end up losing weight.

Have you asked your GP for a referral to a dietitian? Hopefully they’ll be able to come up with a meal plan that works for your health conditions.

Also, can I ask why you are unable to exercise? If you can’t dedicate time for exercise, just increasing your steps and hitting 10k a day can make a huge difference.

I have PCOS and hypothyroidism, so losing weight is really hard. I have absolutely no opportunity to exercise during the day, so I have to rely on my diet to control my weight. My PCOS causes insulin resistance and hypothyroidism drastically reduces my BMR. My doctor won't do anything and has been utterly useless when I've tried to talk to him about it.

I have gradually lowered my calorie intake to 750 and try to keep my carb intake below 30g. I like to eat as clean as possible, so no refined sugar, no dairy (allergy), not much meat and cook vegetable based dishes from scratch.

My daily diet is:
0600hrs - tea with oat milk
0800hrs - protein shake made with water
0950hrs - black coffee
1000hrs - broccoli and white fish
1230hrs - salad with a protein, no dressing
1400hrs - greek coffee and a couple of dates
1700hrs - zoodles and a protein (turkey breast or fish)

But despite reducing my calorie intake, reducing complex carbs and sugar, I am not losing any weight. I am always tired, skin looks awful and my mental health is rock bottom.

I am so sick of hearing that all it takes is a calorie deficit, when I am trying my hardest to reduce my calorie intake and it's still not working.

I am a single parent (during the week, my husband works away from home) working full time and doing an MPhil, there's not even 10 minutes that I can take to go for a walk. I wake up at 0550hrs and go to bed at 0200hrs. I have spoken to my doctor at length and asked for second opinions but all I get told is that I should eat a keto based diet and keep taking my medication. It is hugely frustrating and frankly depressing. I was hoping that maybe there's somebody on here that may have experience of PCOS and/or hypothyroidism that could suggest something. There's lots of stuff online about either PCOS or hypothyroidism but very little of dealing with both conditions at the same time.

Is that just an example day - I hope it isn't the same every day?
No wonder you are tired - that is a very limited diet.
What is the reasoning behind such a low carb count? I would be ditching the protein shake for something else (maybe porridge then you wouldn't need another meal 2 hours later) and try adding in something like sweet potatoes, chickpeas to some of the meals.
I would also swap the dates for a different fruit. 2 dates is 130 calories - that is a huge chunk of your very limited quota - and they are very high sugar.
According to the U.S. Department of Agriculture , a single medjool date contains 66.5 calories and provides: 18 grams (g) of carbohydrates. 16 g of sugars (mainly in the forms of glucose and fructose). Your 2 dates are exceeding the 30g carb on their own.

I've done that and was met with "just keep taking your thyroid medication and reduce carbs and sugar"
My BMR must be around 500 and there's no chance of reducing my calories lower than that.

That is not sustainable - people need more sleep than that, its less than 4 hours. If you get a lunch break you could go for a walk, if you are in an office you can take the stairs between meetings. How old are your children - are they at school/nursery? How much more do you have to go on the MPhil as this schedule is doing you harm.

500?! How on Earth did you work that out?

I work through my lunch break, everybody does here. I work for the NHS so we are hugely under pressure at the moment. I do try and take the stairs as I am on the top floor, but I will admit that sometimes I just take the lift to avoid the brain power of getting my ass up the stairs. My daughter is in school. I drop off her off at breakfast club at 0800hrs (its a 40 min drive to her school) then 20 minute drive to my work. I've not long started my MPhil, but by the time I've done dinner, homework, clean up, get ready for the next day, sort out food and sit down to study, it's already 2230hrs. I agree, it's far from ideal but I don't know when else I would do it.

I am a single parent (during the week, my husband works away from home) working full time and doing an MPhil, there's not even 10 minutes that I can take to go for a walk. I wake up at 0550hrs and go to bed at 0200hrs. I have spoken to my doctor at length and asked for second opinions but all I get told is that I should eat a keto based diet and keep taking my medication. It is hugely frustrating and frankly depressing. I was hoping that maybe there's somebody on here that may have experience of PCOS and/or hypothyroidism that could suggest something. There's lots of stuff online about either PCOS or hypothyroidism but very little of dealing with both conditions at the same time.

I have PCOS and hypothyroidism, so losing weight is really hard. I have absolutely no opportunity to exercise during the day, so I have to rely on my diet to control my weight. My PCOS causes insulin resistance and hypothyroidism drastically reduces my BMR. My doctor won't do anything and has been utterly useless when I've tried to talk to him about it.

Something is going to have to give - otherwise it will be you collapsing with exhaustion. If your daughter is at primary level then homework shouldn't be that heavy (unless it is one of those stupid 'make a model' projects). Maybe drop the cleaning level expectation - leave washing etc till the weekend when your husband can help. By yr6 my daughter was in bed by 9, earlier when she was younger, and I was often in bed by 10.