Spangles, it isn't selfish to want/expect working conditions that aren't painful and/or harmful. Your employer is legally obliged to provide you with a safe working environment. It's not the cherry on top, it's not an option or a choice, it's the absolute minimum legal requirement.
And because you're disabled, you/your employer can get financial help through the Access to Work scheme, which means that your school won't be out of pocket, because they claim the money back from central government.
You apply, they contact your employer, they send an assessor who will do an ergonomic assessment. For a chair, they will measure your arms and legs, etc, and then you will get a report that gives information about their recommendations, which would include the supplier of the chair (so you or the school administrators don't have to faff about trying to to source special chairs). It would be bespoke and made for your measurements. The employer pays for it, then claims the money back from the Access to Work scheme, so it doesn't (permanently) come out of their budget.
Get help at work, including an Access to Work grant, if you have a disability or health condition - eligibility, how to apply.
www.gov.uk
It can take a few weeks, but in the meantime, ask the school to get someone to transport one of your current special chairs to you.
You're not being a nuisance, you're not being a selfish cunt. You need a chair in order to be able to do your job wfh. Your chair is broken. The temporary fix is get someone to transport a chair to you. The longer term fix is to get a chair through the Access to Work scheme. (If one of your school chairs is transported to yours, you will arguably need one in the school. I don't know what the situation would be if you asked for one to work from home (you could ask the assessor?) on the grounds that you do a lot of working from home as a teacher, marking and lesson plans, etc? So the Access to Work scheme might even provide two chairs, I don't know, would be worth asking.
And you wouldn't need to go to Harley Street or see any doctors or any such nonsense, and it wouldn't cost your employer any money, because they'd be reimbursed, although they would have to do a little bit of paperwork.
You're not being selfish if you don't want to be in pain and if you don't want to further fuck up your already fucked up knees. You need those knees for another 40 or 50 or more years. It's not selfish to want to look after and preserve them.
And again, you're not asking for extra special treatment, your employer is legally required to provide you with a safe working environment as a bare minimum, and where a staff member is disabled that might involve a special ergonomically-designed chair, or Dragon software or a big screen and software for people with a visual impairment, a BSL interpreter for meetings, or whatever.
I've had colleagues who've needed chairs through this scheme because of bad backs who've got them.
Put it this way, if a colleague of yours had a visual impairment and needed a big screen and also software that enabled them to zoom in and blow up the size of text and also had a text to speech facility, would you call them a selfish cunt for asking the school to provide them with the tools they needed to do their job? Or would you think it was fair enough? I don't think any of my colleagues raised any eyebrows or called him a selfish cunt when a new starter at a previous job needed a big screen and special software, it was just a matter of Okay, he can't do hot-desking, because that's his screen and that's his computer set up at that workstation. And everyone continues as normal.
Reasonable adjustments in the workplace, providing us with the equipment and tools we need to do our jobs, should be normalised as something that employers should provide as a matter of course, not least because they are legally obliged to make those reasonable adjustments to enable disabled employees to do their jobs.
Disabled people have a right to their place in society and in the workplace.
Please don't undermine yourself (and other disabled people, by extension) by thinking that you're a selfish cunt for wanting the same access to work as non-disabled people, because a special chair isn't about special treatment in the sense of getting one over your employer or colleagues, it's about equitable treatment, it's about ensuring disabled people get to take up their rightful place in the workplace and society.*
[*Obviously, not every disabled person can work, and that doesn't make them less than.]