I think that the advice is to use pads until the post birth bleeding stops. (over 6 weeks in my case, I was not happy!).
I didn't have a period for about 8-9 months after the birth, then used mooncup and occasionally tampons.
Not sure if a mooncup is ok after the birth. Someone like Franny would know. Tampons afaik are definitely not ok for those first few weeks. (high risk of infection)