Religious beliefs are protected by law to the extent that those believers
should not be excluded from mainstream society, by making participation conditional on abandoning their core religious beliefs.
As a mixed race Arab woman, I know many women of faith would quietly - no Twitter storm - just drop out of sport if forced to share changing facilities, pools etc with men.
This has obvious consequences for the health of the BAME population.
Of course, many of the majority female population may have suffered trauma, or simply feel it too dangerous or uncomfortable, to share with men.
They also need separate facilities.
All women have the same right to continue to participate in mainstream society and their needs should not be swept away to satisfy the demands of a small subgroup of men.
I would support any campaign by transwomen to have separate facilities for them, so they can also feel safe.
btw, I have read comments online urging men to identify as women, precisely as a strategy to exclude Muslims from their local gym or pool.