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A car of ours has just failed the first WOF since we bought it for a couple of reasons, including mismatched tyres. It passed its pre-purchase inspection at the time of purchase. What should I do if the registered dealer doesn't want to address this issue ?

Assuming the mis-match is a legitimate WoF fail (ie; different size, construction/carcase type, ply rating or load rating by more than 2), the dealer has sold you a vehicle which is not fit for purpose and should not have had a WoF in the first place. Therefore, the dealer should rectify the problem by supplying a suitable replacement tyre.
If the dealer will not address the issue, you can make a complaint via Consumer Affairs.

Thanks for this advice. The reason for the WOF failure was the tread pattern not being the same. One rear tyre has an asymmetric tread pattern and the other one didn't. The tyres are the same in all other respects.

OK, this is definitely a WoF fail and should have failed at the time the car was sold to you. Are you sure this tyre wasn't the spare at the time the last WoF was done, as it seems highly unusual that a WoF inspector would have missed that?