A Muslim friend and I attended the same breastfeeding support group. She covered completely including the veil. Our group was woman only, so like the girls in EdgeOfACoin’s school she was able to relax and uncover.
I don’t want to get deeply into the proposed law in France, which personally I would never support, but it doesn’t surprise me since French policy has long been if you want to live in France then make efforts to assimilate into the French way of life coupled with strong values of secularism in public life.
Anyway back to the UK without explicitly single sex spaces such as toilets (in school and elsewhere), sports sessions, breastfeeding and other groups there is a very strong risk of cutting women of diverse backgrounds off from each other. Women of particular religious backgrounds who need single sex spaces will likely be able to form them for themselves by privately hiring sports facilities etc which will meet some of their needs but they wouldn’t any longer have the opportunity to meet women from other cultural or religious or non religious backgrounds, then there will be many women who don’t have such connections who just rely on a small social group of friends meeting at each other’s houses. This agenda of removing single sex spaces is likely to prove divisive and isolating for a very large number of women.
To return to the OP I believe that the use of gender neutral language in women’s services is not actually driven by the needs of service users at all but rather by an entirely different group who want woman to have a new meaning which is divorced from the meaning most of us have attributed to it from childhood-adult human female. This agenda rejects biology and concrete definitions and instead proposes that the meaning of the word woman is in the mind of the speaker alone, it cannot be pinned down at all and to try and pin it to biology is bigotry or perhaps blasphemy.
The message of gender neutral language in women’s services is - women you do not deserve words to describe yourselves, you must know your place, others come before you, you are not worthy of a name of your own, your existence is offensive so lower your eyes and your voices, your bodies and bodily functions are disturbing and distressing so do not discuss them.