You don't need BPS accreditation to be a clinical psychologist. The accreditation you need is HCPC. This is the case for any of the practitioner psychologist titles. Psychologist is not a protected title, so you could "be a psychologist" without any accreditation.

What if my masters is not BPS accredited?

Graduates who have not taken an accredited programme in psychology may undertake a conversion course. ... They typically offer their courses in more specialist areas that may offer interesting further study opportunities for psychology graduates.

Is BPS accreditation necessary?

Taking a BPS accredited course is necessary if you wish to pursue a career in psychology or in the wider psychological workforce. This page will help you find an accredited course and learn more about becoming a psychologist.