I'm afraid I think you need to be really blunt with your Mum OP. I know that's hard to do, because you obviously love her, and find it difficult to say no to her, but she is being really selfish. I think you need to tell her that you've hurt your shoulder, and you've seen your GP who has told you that you must NEVER lift anything heavy again, as you have already done serious damage. Tell her that unless she agrees to having an OT assessment, and getting the aids she needs, like a suitable wheelchair and having a hoist fitted to get it in and out of the car, she's going to become housebound, unless of course she wants to start working on improving her walking, as her doctor has told her she should be doing.
If by any chance she tries to talk you into continuing to lift the chair, or working on your shoulder and arm muscles to make them stronger so that you can still help her, you can then tell her that if SHE would only work on making her own muscles stronger, she wouldn't necessarily even need a wheelchair!
Is money for another wheelchair an issue OP, as you may not be aware, but you can actually get a grant from the NHS toward a wheelchair, so that might be worth looking into if money's tight? Also, I may have missed it, but can I ask how old your Mum is, and what exactly her mobility problems are?