The situation with NHS services truly sucks. Especially CAMHS at the moment. They are under resourced and overwhelmed with increasing demand, and the government still just pays lip service to supporting MH services. Don't get me started on Jeremy Hunt!
I'm not sure why the psychologist hasn't responded to your messages, there might be a legitimate reason. But I wondered why prior to that there was mention about fortnightly sessions? There's no right or wrong frequency but it's typically weekly, especially for CBT and when someone is starting out (might go to fortnightly later on in therapy for a variety of reasons). I'd be curious to know what the rationale was for this.
CBT is the recommended NICE treatment for anxiety (along with meds if required) but there is sometimes a reluctance to prescribe medication to children. CBT is also one of the main recommended psychological approaches for PTSD. That doesn't mean other approaches won't work or be more suited, but it's where most of the evidence has been collected.
A pp was correct in as much as we don't have all the answers. Brains are complicated things that we don't fully understand. However we do have some good and well researched approaches that help a lot of people, including youngsters. So please don't give up hope that things can change. I know it's really really hard!
If you need to look for a different psychologist, try the British Psychological Society website. Some psychologists who do a lot of CBT will be registered with BABCP as well (they have a website), but not being on there doesn't mean they don't or can't do good CBT. The main thing with a clinical psychologist (or counselling psychologist) is that they will take time to formulate and understand the situation first rather than leaping in with an intervention that isn't suitable for the issues at hand. Anxiety has heaps of different causes and different things help different people depending what the causes are.
Wish you and your daughter all the best.