If it is Infant Class Size ie intake is full it is unlikely this would win an appeal. Is there any health or social needs category in the admissions criteria, although this relates more to those of the child than parent/carer. Are you on the waiting list for preferred school?

Is this an Infant Class Size (ICS) appeal, ie the school admits in multiples of 15? If so, I very much doubt that you’ll win.

For an ICS appeal, you need to show that there’s been an error which has cost your child their place, or the decision to refuse was so unreasonable (in the legal sense) that it should be reversed.

The panel will be sympathetic to your difficulties in caring for your husband, but I doubt they will agree that they meet the very high threshold to win an ICS appeal. It would be a bad idea to tell the panel that, if your child doesn’t get a place at the preferred school, they’ll miss school. That sounds like blackmail. The panel will not like it. They are likely to take the view that it’s your job as parent to get your child to school and, if the offered school is less than five minutes away, seek help from a childminder or neighbour on days when you can’t take your child yourself.

Is deferring your child’s school start an option? That might give you more time to sort out arrangements for getting your child to and from school.

Sorry not to be more encouraging.

Do you have a waiting list position ? Hopefully you'll be very high and there's some movement before September.

It's unlikely your circumstances would win an appeal - what class is your sister's dc in? If a higher class what arrangements would you have in place when her dc leave? Agree with other posters, don't say by not being in the same school as your sister's dc your child might miss school - it's not a good enough reason and the tone of "You'll cause this" won't help you.

It might be worth looking at nurseries, childminders etc that do school drop offs and pick ups, and forging links with other parents when your dc starts to build an I formal support network for drop offs and pick ups.

Some schools do have a category for wider family social/medical criteria in their admissions ordering, but it sounds as if either that's not the case at the school in question, you didn't gather evidence to apply on this basis, or if you did, there were still children whose applications met higher criteria. The school will understand your difficulties, I'm sure, but unless a mistake has been made they won't be allowed to admit your child.
This doesn't mean you shouldn't try - mistakes do happen. But realistically, your best chance of getting into the school is to be on its waiting list. People move, or change their minds, or simply don't show up in September. Don't give up hope, even if your appeal is unsuccessful.

If the school is only 5 minutes away I'd say there is a pretty high chance that you have a close neighbour whose child also goes there who would be happy to take your child to school on the odd day that you can't make it.

Are the start and end times of the two schools exactly the same? Your sister might even be able to manage to get round both with them being so close.

Non of the above will win an appeal. It just won’t be believable that your dh can’t be left alone for school runs. Even if that were the case you would be expected to make other arrangements.