I think the problem I have with the OP is the fact that she doesn't appear to be asking the right questions and to the right people.
Its the fact that she's sought legal help BEFORE perusing a complaint within the system and seemingly even giving the doctor she has a problem with the right to reply.
Step one is asking the relevant parties what they think happened. She seems to have done this with the private hospital and got brushed off (when I don't believe she has had a satisfactory response from them anyway) and taken this at face value without too much question. The fact she didn't with the other medical party leaves me bewildered.
Hence my question about what she is seeking to achieve.
If making a complain about a GP, the first port of call is to discuss with them first to establish if there is any misunderstanding or explanation you might not have been aware of. If someone isn't comfortable with that, then that's what PALS are for. That's the point of investigation - before external involvement.
Its not a call to some legal firm who want to make money from your misfortune. That does not improve health care nor does it deal with failings in procedures. All it does is put extra pressure on all doctors, not just those bad ones. It just encourages defensive medicine rather than good or appropriate medicine whilst is fundamentally different in nature. The trouble is that early and unnecessary legal involvement discourages transparency rather than encouraging a climate of openness because of the implications.
If you want to change things and improve procedure and prevent similar mistakes in future then working WITH doctors rather than against them is more productive. This is the way you should approach complaints with the NHS - in the most constructive matter possible, This means engaging with them and having a two way discussion over the case. This can also have psychological beneficial affects for someone who has suffered a medical error or mistake as it gives you back the control and makes you feel something good has been achieved despite the problem.
Legal redress should be the last resort when all other avenues have failed simply because their remit very different in nature to complaining via other channels.
Legal action DOES have its place and its use within malpractice, but it needs to be used appropriately otherwise it has the effect of being part of the problem rather than the solution.
As I say, this is why I raise my eye brows to the OP, especially when I read what she's said and find the advice she seems to have been given by a law firm as somewhat dubious in nature.
What she wants to achieve here is important. Thinking that a legal investigation is the best way forward, at this point, is something that I do not think she has fully thought through. As others have said, the burden of proof is different, as is the stress it puts on everyone involved as agendas are different.
Finding out what went wrong, is about being able to ask the right questions to the right people, in the right way. Stopping and thinking about your end goal is a crucial part of that.