We spend less than £130 on gas, electricity and water combined in a small house, though there are only three of us. We are not cold or scrimping and not actively trying not to use gas, electric or water! Phones can be very cheap - the cheapest GiffGaff sim only plan is something like a fiver a month. Nobody needs the latest model and even those are sometimes available at discounts. My daughter had a new (but not the latest) phone a couple of years ago for less than £200. It does everything any other phone does.
You're right about internet and petrol (if they have a car). But if they are in London and looking at a sensible geographic area in relation to where they live travel for the children is free on buses and heavily discounted on the tube. The parents may well be able to work at home at least part of the time
Internet can be shopped around for. We currently have fast internet for £26 a month. Yes, it took a bit of negotiating but well worth the time. I think four (free but longish) phone calls in all.
I didn't think about uniform but there are usually secondhand items to be had. I think I spent less than £500 over DD's entire secondary school career, though she did not need as much as some other schools. Even at three times the price per child that sounds doable with some planning. Not sure how much lunches cost as DD's were included at her school.
Nobody went on all the residential trips. I don't think it would have been remarked upon or even noticed if someone hadn't gone on any. I appreciate the theatre trips might be compulsory but these were heavily discounted. I think around £25 per trip for us which I personally think is an absolute bargain for West End theatre, the National, the Royal Court etc.
I don't deny that it might be really difficult and involve a lot of planning and budgeting, and I am not sure I would want to do that if it meant really not having enough for some treats and stuff like music lessons etc (though music scholarships normally involve free lessons IME). In OP's shoes I'd be looking at the superselective London grammars if she is in striking distance for any of them.