Welcome to the forum.
They should have had Attendance Allowance years ago! Get the forms completed and returned asap.
The more help they get, the longer they can stay in their own home, but do NOT make any promises about “never put you in a home”.
It would help to have a bit more information to give you best advice. Sorry, questions are the quickest way, don’t answer any you don’t want to, but there is a purpose behind each of them.
Mum is seriously disabled. Does she have a hospital bed to make caring for her easier?
Has an occupational therapist ever visited to talk about ways to make caring easier?
Do they own or rent their home?
How old are you?
Do you have a home of your own?
Do they have over £46,000 in savings? (the limit for Social Services care). Just Yes or No for an answer.
Do you have any brothers or sisters? Power of Attorney?
The line “I don’t want anyone else in the house” is so common, but they have to accept that they can only manage now with a lot of care.
All four of our parents lived through the 2nd WW, the “I can manage” principle was common, even when they clearly couldn’t!
It’s not up to them to tell you what to do, but you choose what you do, or don’t do. Then arrange for additional help as required. My own mum was adamant she didn’t want any strangers, but once she was left with no other option, when I was having cancer surgery, she finally accepted help.
Once she got used to the carers, she enjoyed hearing about their lives and their children, welcome female company.
It wouldn’t be unreasonable to use their AA to hire someone to do basic cleaning and meal preparation, you are their son, not their slave. If she came twice a week, initially working under your guidance, you could gradually pop out for longer and longer periods, knowing they are safe.
Do they both have Lifeline pendants to call for help?
An accessible bathroom?
Do they have an easy to maintain garden, and a gardener?
Finally, last but not least, has her doctor ever mentioned NHS Continuing Healthcare to you?