To ask whether you have to prompt your SO to pick up the childcare?

[Anybelle]

On the assumption it's normally you who does the majority of the childcare whilst at home, do you have to prompt your DP/DH to take over looking after the child/ren when you're both off so that you can go and get things done/have some time to yourself, or does your SO use their initiative to give you this opportunity without you asking?

YABU - ask, other people aren't mind readers
YANBU - it's normal to assume someone who doesn't get much time to themselves needs some down time

I’m assuming your DC are babies/toddlers? At that stage - honestly yes I would specifically ask/tell but by the same token so did DH. So neither of us would plan something child free at the weekend without checking with the other if they were ok to be in sole charge of the kids. I think it was similar in the evenings - I would say ‘DH I’m going for a shower now so you’re in charge of the kids’ but if he’d been with them all day I wouldn’t get home and assume he needed time alone.
I’m not saying this was a perfect set up but just how it worked out for us.

Depends on a context.
If you are all in the same room and you need to step out, then just saying "I am going to do xyz" and then going ahead with it will surely mean that an adult left in a room is going to keep an eye on a minor.
If you are in separate rooms and you need to do something, then going to where your partner is and saying "here is the baby, I am going to take a shower" seems reasonable as he is not a mind-reader.
Similarly "I am booking a manicure, are you able to pick the kids from school on the 20th or is another day better for you?" seems like a normal thing to do.
When the kids are older, "go find daddy" also works fine.