"Counsellor" and "therapist" are not protected titles. They are often used interchangeably. A "therapist" could mean a lot of different types of training, in NHS world (and outside of it). Their core training could be counselling/psychotherapy or CBT, or they might be a nurse or OT with CBT, DBT or EMDR training. They call PWPs "therapists" in the NHS, and those are usually wee psychology grads with a 1 year postgrad certificate course in doing shit manualised CBT, not therapists at all really.
A "therapist" could also denote a clinical, counselling, educational or health psychologist with additional therapeutic modality training like CBT, EMDR, or psychodynamic therapy. Or if we're talking CAMHS, quite a few have systemic/family training, which people can go into from backgrounds like psychology, counselling/psychotherapy, social work and teaching. CAMHS may also have art psychotherapists and play therapists as well as CYP counsellors/CBT therapists.
Counsellors/psychotherapists could have anything from a Level 4 diploma up to a Masters or doctorate. Systemic/family therapy training is generally 3-4 years and you come out with a PGDip or Masters. Art psychotherapy varies, as does play therapy.
CBT therapists would usually do a postgrad diploma while employed in the NHS for their practicum. But not always - nurses, OTs and psychologists who convert might do fast track training.
Psychologists have a doctorate. But that doesn't always mean they are qualified to practise any and all psychological therapies, it depends what their core training included. I trained in educational psychology originally, and converted from working for the LEA to training to do systemic therapy in CAMHS. At the time it was NHS funded. Not many funded places available these days though.
Some types of therapy require additional post-qualification training, for example EMDR, you have to have a certain amount of experience and be accredited by BACP, BABCP, UKCP etc if your core training is CBT or counselling/psychotherapy and you want to train in EMDR.