Mothers partner. Being upfront saves any awkwardness later. The know the exact relationship. He doesn't get called 'dad' by a member of staff, he doesn't have PR (I'm presuming he's not married to you as you would have said your husband) so the doctors know they need someone with PR to make medical decisions. The school will see it all the time, there won't be any judgment, and it saves feeling like you have to explain or give any sort of details later on because of someone misunderstanding/mis-titling or just generally embarrassment. (Obviously you don't owe anyone any details but as soon am I'm tripped up I just end up over explaining 🙈) (But that's just my two cent).
Step dad/mom if you're married. Actually, I'd probably ask partner if it's ok if you put step dad/mom anyway.
The other choices are 'Mom's partner' or 'Mom's boy/girlfriend' arent they really. And though there's nothing wrong with that (I'm a mom and I call my partner my boyfriend), it's not really as official sounding (for want of a better word).
I think I'd talk about it with my partner first and see what they thought. Then if the kids were old enough I'd probably mention it to them in case it ever comes up at school.
thatwomanwho - we've always put step-dad on paperwork. People still seem to assume 'dad' even when we've told them in person from the start! Although at school and clubs it is possibly because they have seen DH regularly and DS1's dad less than once a year!