"Our local Maternity Voices Partnership now refer to 'birthing people' in correspondence, social media etc."
The irony! "Maternity Voices"!
Odd example of usage in the definition of maternity below but otherwise makes the meaning plain:
maternity | məˈtəːnɪti |
noun [mass noun]
motherhood: she is not a woman with an interest in maternity.
• [usually as modifier] the period during pregnancy and shortly after childbirth: maternity clothes.
ORIGIN
early 17th century: from French maternité, from Latin maternus, from mater ‘mother’.
I agree with previous posters - I would assume "Birthing person" to mean anyone other than the mother involved with the birth.
This sort of bizarre language is confusing enough if English is your first language and you have a reasonable level of literacy. The aim should be to communicate effectively and think about "hard to reach" groups - not alienate and confuse your target audience.
"How to write medical information in plain English" - Plain English Campaign
Ten tips for clearer writing
1. Think of your audience, not yourself.
Don't try to impress people by using your language to show off: keep it as straightforward as possible. Imagine you are speaking to someone, and write in that more relaxed way.
8. Use everyday words.
Big words, foreign phrases, bursts of Latin and so on usually confuse people. Consequently, it is a sine qua non of plain English not to write too polysyllabically! So, for plain English, use everyday words.
Download guide: www.plainenglish.co.uk/files/medicalguide.pdf
Plain English Awards 2019 include NHS Scotland ‘Ready Steady Baby!’
www.plainenglish.co.uk/campaigning/awards/2019-awards/plain-english-awards.html
Maybe you could suggest that the local "Maternity Voices Partnership" prepare info to a standard that might result in them winning an award from the Plain English Campaign?
For a bit of light relief - the Plain English Campaign "Gobbledygook Generator" 
www.plainenglish.co.uk/gobbledygook-generator.html