I understand the stress and guilt, OP. For better or worse, I was fired 3 months after my child was diagnosed with a critical illness, as they couldn't afford to keep paying someone who needed to keep taking time off. I honestly don't know how anyone manages to keep any sort of job with long term caring responsibilities, unless they can work from home, have a highly qualified and trustworthy nanny and/or a partner to share some of the load. Even then, a nanny can hardly take over the responsibility for organising medical appointments or taking a child to a hospital. I guess a personal PA could do the admin side of things. Are you a single parent too? That does make it harder.
There is some generic info on your entitlements on .gov.uk and carersuk:
https://www.gov.uk/time-off-for-dependants
https://www.carersuk.org/
From experience, I would try to hang onto your job as long as humanly possibe. Even if you think you're doing a crap job, just hang in there and do the best you can. Life is infinitely worse with less money coming in.
In the meantime, if you haven't already, apply for DLA for your child, carers allowance for yourself and see if there is any other help you may be eligible for e.g. there are a lot of charities out there who offer a volunteer driver service to pick you and your child up, take you to hospital appointments and drive you home again. A small thing, but means you don't have to worry about parking or paying for a taxi etc.
Look up charities that were set up to help people manage conditions like that of your child. Their websites will usually have links to their own services as well as other services that may offer something different. Ask your Dr/the hospital for referrals for benefits advice and to any charities or support groups that may have more info.
If you're eligible, don't delay making the DLA claim. They're taking 6 months to make a decision, by which time, the choice to keep your job or not may have been taken out of your hands.
Good luck, OP. I hope you're able to find some support and others in a similar position can offer more advice.