Previous convictions should most certainly be admissible, provided that the prejudice does not outweigh the probative value. For example, if the defendant is on trial for murder, an unrelated drug offense committed twenty years ago serves only to prejudice the jury, and adds nothing to the case itself.
I can understand how it could be relevant to disclose previous convictions if they were acutely similar to the current case being tried; for example in the case of a rape it would be good sense to disclose if the defendant had a previous conviction for rape under similar circumstances (rapists often have something distinctive so you can generally tell if two rapes were committed by the same person). Other than in such cases, I do not see how it could be probative in a completely unrelated case. For example if somebody has a conviction for GBH I do not see how it could be deemed probative evidence in a separate case for murder. I think that under many circumstances previous convictions do not show a propensity and so are not probative evidence. The only relevance I can see is if there were parties involved in both of the cases other than the defendant.
In many cases I would say that bad character references simply tilt the balance away from the defendant by giving the jury preconceptions. I think to a great extent it can undermine the standard of proof of innocent until proven guilty by prejudicing a jury.
I think the criteria set by the CJA 2003 aren't bad criteria if any form of bad character references are to be admissible though I am less sure about subsections g. I think the explanation of c in section 102 of the Act is satisfactory for the most part to allow that.
On a general principle I would prefer for previous convictions not to be disclosed, but I recognise that there are necessary exceptions.