I wasn't aware of this, so hopefully it's helpful for other people that don't know.
By law, council vehicle tests centres must be open to the general public. They usually only carry out tests and not repairs, so there's no incentive for mechanics to find faults that don't exist.
They charge a maximum of £54.85 (some have posted their local charges around £30), but hopefully this would be offset against fewer repairs. Private companies may run cheaper special offer MOT deals, but it could be in their interest to exaggerate any repairs that may need doing.
The council test centre might obviously still say that a repair is needed if they come across a problem when doing the MOT, but at least you know it's genuine.
Full list included in first post.
Thank you to those who have posted additional centres, all added to the list.
(Sources: MSE and other sites)