Hi Frewink,
Unfortunately Skeptical is right. You have to have a Psych degree accredited by the BPS (as well as several years relevant experience) to be considered for any of the postgrad training courses that qualify you to be a psychologist. Such as the Doctorate in Clinical Psych, or Educational Psychology, Health Psychology, the Psychotherapist training etc etc!
I'm sure there are some Masters level courses that would consider you, but even if you had a Masters such as Research Design Methods (many aspiring psychologist undertake a masters to boost their chances of getting on the Doctoral level training) you would still be ineligible for training as your Bsc Hons Psych degree is not BPS accredited.
However, you may be able to undertake a conversion course masters to gain BPS accreditation with a uni that runs the BSc Hon Psych degree accredited by the BPS. However this will incur more cost for you and could be avoided if you did a BPS accredited degree in the first place. People generally do the conversion course if they have undertaken another degree or some joint honours degrees and want to gain a Bsc Hons in Psych in order to pursue a career as a psychologist.
So I am afraid BPS accreditation is really important.