Here's how I would cut back:
mortgage & utilities - 1198
car loan - 216
Terrible plan, but I guess you're stuck in it now until the term is up? Or can you sell it and pay off the loan then get a cheaper car? There are loads of used options on Cazoo for about £100-£130 a month and it's a false economy to think a 5-year-old car needs much more fixing than a brand new one these days.
loan repayment - 175
credit card - 100
kids sports/hobbies - 415
You've said these are non-negotiable.
car insurance - 90
That is £1080 a year. I arrived in the UK with 0 years no claims and bought a convertible and paid less than this in insurance. Get on compare the market and see if you can save something here, renewal quotes are pretty much the most expensive way to get insurance these days and you can cancel any time during your term (usually costs about £40 which might save you £100s)! I'd be surprised if you can't get it down to about £50 unless you drive a Porsche around like a maniac hitting everything you see.
car tax - 14
sky tv - 10
Netflix - 10.99
Cancel it. You don't need this and sky, and Netflix is on a month-by-month basis.
mobile phones (tesco family sim only deal) 50
Cancel this and go with Giffgaff, this isn't the deal you think it is. Giffgaff is PAYG but you can set it to recur each month and it is really the cheapest thing going.
savings - 70
There is absolutely no point having savings while you are paying off debt because the interest on the debt is higher than the interest on the savings. Put this towards the credit card so you can be free of it sooner.
groceries - 500
Three kids and one adult? Cut back. Tesco value pasta and pesto once a week. Buy tinned instead of fresh veg. Go down one level of snack (or right to the cheapest if you're up for it). Stop buying any alcohol at all until you're not haemorrhaging money. Go for frozen meat, not fresh, or substitute meat for lentils or peanuts. Use the time you thought of spending on the hypothetical second job cooking from scratch and eliminate as many UPFs as possible, as UPFs are more expensive than cooking from (mostly) scratch if you make sensible choices.
Car fuel - 400
I assume this is non-negotiable as I'm assuming you're not just driving around for the fun of it and where I live, public transport is much more expensive than driving.
iphone storage 2.99
Upload all your stuff onto a computer and cancel this. Every time yours/your kids' phones get full, upload to the computer again. We all used to do this before the cloud turned up, it's not a hardship.
nails & eyebrows - 30
You cannot afford this right now. A bottle of (nice) nail varnish is about £10. A pair of tweezers is £3. A brow pencil is £3. For the cost of not even one month of this you can buy the stuff to do your own and follow a free Youtube brow tutorial. And you can spend the time you are wasting sitting in a salon doing something more productive like earning more money.
spending money - 400
Try £200.
birthdays & Christmas savings - 200
Bring back the real meaning of Christmas/birthdays and make these celebrations low-key for the next 12 months while you stabilise. The kids can't have expensive hobbies AND big birthdays and if the hobbies are non negotiable then the celebration fund needs a major trim. We spend about £100 on Christmas celebrating and £50-100 per child on gifts. We spend about £50-100 on birthdays. You're not going to give them a nice memory if debt collectors turn up and clear your house of everything you own.
Total savings here: about £500. Leaving only £300 to find which is much less awful.
More suggestions to bring in money:
Go through your attic and sell everything you can find. Do the same with your garden shed. I sold all my (cheap plastic) plant pots for about £5 each!
Any teens over 16 in the house? Ask your teens to get part time jobs and charge them board of about £30 a month. They don't need to have set hours. Car washing. Dog walking. Babysitting. Dogsitting. Cleaning. All things I did to bring in money as a 14-year-old onwards.
Rent out your driveway to people on Justpark.
Rent out any spare rooms in your house on Air BNB (you could put a bed in your dining room and eat in the living room or kitchen). You could choose to only do this on weekends to have the house to yourself during the week.
You said you're paying for iphone storage. This is a controversial one. Sell your iphones and buy cheap Androids. Even older iphones hold a higher resale value than android phones so you could raise a bit of cash doing this but it's not for everyone.
Can you downsize at all? Or take out an equity release loan on your house and pay off your debts completely? This is basically adding your debts onto your current mortgage and usually gets you a much better interest rate than loan interest. Just make absolutely sure you pay off ALL your credit with the money.
I hope some of this is helpful. We were in this situation in January and I contemplated ending it all so I know how terrifying this is. I ended up having to sell loads of our stuff and totally change job to something that brought in enough to get us out of the mess we were in.