I'm not a psychologist or a psychology graduate but I used to deal with applications and enquiries from people who were, for a very competitive MSc in psychology. Most of them wanted to proceed in due course to the doctorate in clinical psychology. The majority had a good first degree in psychology. Some had a good degree in something else plus a good result from a conversion course. The key thing in both cases was that the psychology qualification needed to be approved by the British Psychological Society to give Graduate Basis for Registration (that might not be quite the right term, but it's something like that - it indicates that the graduate has covered all the subjects the BPS considers a chartered psychologist should know about as a foundation for further study/training). Without GBR you can't proceed to postgraduate study (also accredited by the BPS) to become a chartered psychologist.
For clinical psychology, the typical applicant seemed to have a CV like this:
Good first degree and GBR
Relevant work experience, often starting with voluntary work during the first degree
Master's in a relevant subject (not essential, but strongly recommended as it gives training in research methods and academic writing)
Ph.D. (not by any means essential, but a surprisingly large number of aspiring clinical psychologists did a Ph.D. first)
It often took years and years, a decade or more even, from completing the first degree to getting onto the DClinPsy, and many fell by the wayside and ended up doing something else. Hardly any of our students got straight onto the doctorate from our MSc, but quite a few managed it within the first few years after that.
Other options our students looked at were proceeding to training as forensic, health or counselling psychologists.
The BPS website used to have a lot of information on it. Worth looking at that.