Hi
If your son is 16 and has a MH issue, a Clinical Psychologist would be appropriate.
This NHS page explains a bit about the role : www.healthcareers.nhs.uk/explore-roles/psychological-therapies/roles/clinical-psychologist
All Clinical Psychologists have to be registered with the HCPC by law. They can also choose to be registered with the BPS (but don't have to).
Chartered isn't as important. Just means they have experience and are a member of the BPS (lots of Clinical Psychologists aren't)
Psychoanalysts have a different type of training and don't legally have to be registered with any particular body.
Anyone can call themselves a Psychologist. It's not a legally protected title and you can't be sure what their training has been.
You're likely to find Clinical Psychologists charge the most - they've done a minimum of six years at Uni - an undergraduate degree and a Doctorate (unless they qualified a long time ago). And most have done more time at Uni than this and have a number of years experience as well.
It's confusing - you're not alone in finding it confusing.
If you decide on a Clinical Psychologist, I would check they are registered on the HCPC website. Here is the address: www.hcpc-uk.org/check-the-register/
I hope you and your family are able to find the support you need.