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It should be shared. His time is not less valuable than yours because you have a family. You sound like the selfish one.
If you can't do physical care for the same number of hours, then what about offering to do other things so it's not *all* on your BIL?
For example if you're really good at organising schedules, can you be in charge of making appointments that MIL needs to attend, and booking cabs ahead of time?
If your husband is a great cook, can he 1.5x or double the recipes and freeze the remainders in single serves for his mum? (Get a big packet of takeaway containers from your grocery store so you're not waiting to get your dishes back. They also freeze and stack easily.)
Can one of you drop by to pick up the dirty clothes/washing a few times per week, wash & dry at your home, and return it folded and ready to put away? When you return it, pick up the next load. Have your husband ask his mum what schedule would suit them, eg every 3 days?
Can you pay to get her *some* in-home care so that it's not literally up to all of you? She may be annoyed that her children aren't able to drop everything and care for her, but you all already have lives and you can't necessarily put everything on hold for her. A carer 1-2x days per week will mean you *are* sharing the task of taking care of her, just not with your actual bodies, but you are nonetheless ensuring she is cared for.
Just some ideas! See what works for you and your husband. He's her child so he should have the difficult conversations with his brother and his mother, don't let him throw you under the bus for this. You can of course have your say and be present, but he needs to primarily shoulder their anger/discomfort.
Just suck it up and do it. I was my dads carer just before he died. None of my 4 siblings did a thing. I'm just glad I got that time with him, I didn't care how I got it
Talk to the staff at the hospital and let them know she doesn't have care at home, they will know what to do, you can get short term care for her. But you can't let them discharge her before it's sorted, otherwise nothing will happen.
Maybe the MIL can apply for in home care assistance, or she may have to pay to fill the gap. You might be eligible for the Carers Allowance thru Centrelink.
Gathering from other comments she’s broken her hip. If she’s still in hospital ask to speak with a social worker or discharge nurse. If neither of them are available speak to the unit manager and make it clear you are unable to help at home. MIL will be eligible for home nursing services, make sure you utilities all the services that are available to you. But it must be arranged before discharge or she won’t be eligible. Some patients are eligible to remain in hospital for rehab and only come home once independent. You need to speak to someone involved with her care.
Hire a carer! Speak to community care services in your area like blue care. If bil does not want to do this, which is common especially when it’s a male toileting and showering his own mother there is a huge risk of neglect!
Him being single and working less is irrelevant
Hire a carer for your share, I can’t imagine he would want wash his mum or take her to the toilet.
What are you required to do, exactly?