DDCC We gave gifts to the midwife and her student at the last appointment, when DD was 6 weeks, as a thank you (beyond thank you, of course ) and goodbye. I think either would be appropriate. I am sure even if I'd had something immediately after the birth, I wouldn'tve thought to give it to her but I am sure it would be nice either way.
I didn't even think about the gift after the birth and I wanted to be in a clear state of mind(and the MW as well) when I gave it to her and thanked her.
(My birth was at 130am...hence the foggy brain....)