There are a number of angles from which this is a very serious problem for effective healthcare delivery. Sorry this is going to be a very long one but I hope it's helpful OP.
The first and most visible one is from a public health perspective. Effective public health communication relies on use of language which is easy to understand and accurate. Complex health messaging using technical language (of the 'people with a cervix' kind) is a barrier to the public's understanding of their own health. People with low literacy, or low comprehension skills (which frequently overlaps with general deprivation or protected characteristics such as disability or belonging to a minority culture) are disproportionately affected by this and complex messaging may therefore reinforce health inequalities. This is very common. 42% of the working age population in the UK struggle to understand everyday health information: service-manual.nhs.uk/content/health-literacy
The public-facing language of the NHS is enormously important because it sets the tone and context within which healthcare is delivered and communicates something about the NHS's priorities and values to the public and to the staff on the ground. That is why this is shaping up to be a massive battleground. If the NHS or senior medics can't bear to say the word 'woman' that is a subtle communication of how the health and care system views women's interests and wishes. That is why the weaselly public health and policy language which just happens to completely avoid using the words 'woman' or 'female' is a problem. It has political consequences because it means you are losing language about women from the public sphere and eroding public acknowledgment of particular issues as uniquely affecting women-as-a-class. This is damaging for women because it erodes feminist analysis, as well as the argument for women's inclusion in public life and representation in political decision making - if issues like abortion affect 'people' rather than women, there is no longer necessarily a moral case for including women in decision making about those issues.
Secondly it is critically important that the health service does not lose language to do with women or with physiological sex differences because if you lose the language of these, you have no way to describe the concepts associated with them. You then lose something analytically/conceptually very valuable, because sex is a fundamental biological variable which affects almost every aspect of how medicine is practiced. It affects everything from patient safety (in terms of knowing the right reference values for lab tests and dosages of drugs that are appropriate to the sex of your patient), to diagnosis and treatment (understanding the epidemiology of different diseases and how diseases present and progress in each sex, understanding how pregnancy in women patients affects how you deliver care for them), to delivering care with valid consent (e.g. if a transwoman treats a patient who has asked for a woman clinician, you arguably haven't obtained valid consent). It affects the research which develops better treatments, because you need to understand who you are recruiting to your trials and how different treatments affect outcomes for each sex. It affects the health data which informs research and policy - you have to be able to clearly identify and define women in this data in a stable way to be confident that you are designing good policy that meets their needs. I could go on and on. It is vital that clinicians and policy makers are able to conceptualise, clearly name and act on sex differences when delivering care and designing policy. This may now be at risk.
Fundamentally the point is that if you choose to jettison the language, you will ultimately at some point, in some way end up impacting patient care. Care based on a lie is bad care.
Thirdly as you say there is a respect and dignity angle to it - a publicly funded health system has no business making the correct name for 51% of the public a taboo word, especially in the context of medical issues that only affect women like pregnancy. That is not serving the public. Genderism is a political project and the NHS should be politically impartial . Doesn't mean that you can't be inclusive of/respectful to transgender people but they should not be made an excuse for neglecting respectful treatment of the majority.
You ask for solutions OP, and I'm afraid that this can ultimately only be solved by a grass-roots movement building a visible mass public consensus that mobilises action at political level. This is because the ideological capture in the whole health and care system (NHSE, DHSC, industry, academia, think tanks) is so pervasive and goes so far up that it is not plausible that it will be challenged from within any time soon. Almost all senior health and care leaders at least publicly buy into this language and ideology (most disappointingly of all, even the saintly Chris Whitty), so don't underestimate how far this goes up. But anything you can do to plant a seed with someone, or at least get them not to go publicly all-in on genderism is good, because it will make it easier for them to climb down when the tide of public opinion turns.
But it is really the public scrutiny that will make the difference here. Politicians really, really care about being reelected. I think Janice Turner said something recently, that every politician should be asked at the next election on where they stand on males in teenage girls changing rooms. Forcing the issue in this way and making politicians understand where the public are on this is the way forward.