The point of getting a b12 test done on the NHS is that if she is low, they have no excuse not to give her injections. You're right, tests are often misinterpreted and a lot of tests themselves are totally inadequate and not sophisticated enough to pick up deep-rooted biochemical problems. So even if the b12 levels are considered "normal" which is likely, it doesn't mean it's getting into the cells. Then you know you have to go private for the shots.
Re glutathione, it's to do with its chemistry. Basically being a sulphur compound it can stir up too many metals without chelating them in a safe way. For proper chelation you need alpha lipoic acid which increases intracellular glutathione anyway. You'll need to read Cutler's work Amalgam Illness but obviously you have other more important things to deal with at the moment.
Good luck in finding a better quality home for your mum.