'Birth mum' is a perfectly ok term, but so is 'birthing parent'.
When we were in the hospital, plenty of people didn't realise DP was a woman (she is, she's not trans, before anyone asks). There was no point saying 'she's the birth mum' as they just associated the phrase with adoption.
If the guidance for midwives etc. referred to 'the birthing partner' I'm sorry, but I can see it might help clue people in that the person giving birth may not look stereotypically feminine to you, but they still deserve care.
Btw, @HermioneKipper, I don't know which of the many trans men who've given birth you're talking about, but the trans men I know who have children seem pretty calm and down-to-earth about this. Certainly no interest in 'erasing women'.
I get why people worry about all sorts of issues around women-only spaces, really I do, but sometimes - and this is one of the times - I think mumsnet really shows itself up.