Yanbu, your employer needs to manage cover much better, 2 off at the same time in such a small team for such a long time is completely out of order! Most employers don't allow more than 2 weeks at a time and would not allow leave to be taken at the same time as another staff member if a small organisation/team.
There are also legal requirements when it comes to care, even when it's being provided in a private home, I'm afraid I don't know the current legislation but I'm sure acas would.
As for the grandma, I would say to her something along the lines of you understanding (but NOT apologising for) the stress the situation has caused BUT that it isn't as a result of anything you've done and that your taking time off at short notice was unavoidable as you had TAKEN ILL. And that while you understand, you won't be spoken to disrespectfully or angrily when you've done nothing wrong.
"We don’t get paid when off sick at all" why?! That's illegal! You should be getting ssp at the very least - although I'm shocked to learn you need to be off min 4 days now - am I misremembering that never used to be the case?
Although to be honest any DECENT employer should pay sick pay anyway. Sadly in the current climate it shouldn't really surprise me that such things are disappearing. Are you in a union? If not I would highly recommend you join one ASAP.
Greeborising - NO! Just no! Nobody has the right to take that stress out on someone who has done NOTHING wrong and is not responsible for the stress being caused. Op is an employee not a whipping boy! Your attitude to people in caring professions is completely unrealistic and frankly disgusting! Everyone needs down time and op can't help being ill. To expect carers, Drs, nurses etc to be "on" 24 hours 365 days a year is ludicrous!
And I DO have experience as a carer - both professionally (including as a manager) and personally. Your attitude stinks!