Several people I know who are British, and grew up on the NHS, have very negative opinions regarding the NHS' ability to diagnose illnesses.The consensus from them is that the NHS is great for treating conditions that are obvious (a broken bone, a heart attack), but not very good at diagnosing things that aren't obvious. Of course, the NHS is great when it comes to providing medical care for people who in the US would have no medical care at all.
I have found that when it comes to healthcare in the UK, it is essential to do your own research and take control of your own healthcare and not depend on the NHS to figure out what is wrong with you.
When I met DH, he had been suffering from a medical problem for about 20 years that he was never able to get diagnosed, much less treated for. He had gone from doctor to doctor, who treated him as though his condition didn't exist. I did some internet research and not only found out that he had a real condition - with a name - but at least one internet forum full of people suffering from the same disorder. It wasn't until he arrived at the GP's office carrying a printout from the internet that the GP even considered referring him for further tests.
I spent about three years in the UK suffering from a severe problem that was never correctly diagnosed, despite many visits to the doctor. I did some research on my own, realised that it was likely to be caused by a vitamin deficiency, changed my diet and voila! I seem to be cured. No GP ever asked me about my diet or suggested that I be tested for a vitamin deficiency.
I definitely do not think the OP,s fear is irrational. The best way for her to cope with it is to be as knowledgeable as she can about her own health, so that she doesn't have to be completely dependent on the NHS.