Misdiagnosed lung cancer, how do we make a complaint?

Hi. My mum has been diagnosed with lung cancer and beaten it twice. Just over a year a go she had trouble breathing so we took her to the local A&E and they send her to breathing unit there. After a CT scan and a meeting with a doctor she was told the bad news that it had come back and there's no treatment for her, basically told she was now terminal. After putting her life in order, making a will and saying goodbye to family. A year later after a number of visits to the hospital for checkups has been told that she hasn't got cancer but a lesion on her lung and told they will see her in 12 weeks to just keep an eye on her. I can't contemplate what has been going through her mind but I was so elated by the news. Looking forward to hopefully many more years with her up it would have been nice for the doctor to say sorry he go it wrong. I love the NHS but to  have this hanging over her head for the passed year, can anyone advise on making a complaint. Not for any financial gain just a written apology. 
Daisy