Unfortunately, it depends on your case.
If they accept liability and agreement is possible as to how much your claim is worth, or they are willing to offer you a sum of money you consider acceptable without admitting liability, it could end at any stage.
However I expect the NHS will want to see your medical evidence before making any offers. Your solicitor will probably send you to a medical expert to prepare a report - you need to ask your solicitor timescales for this.
If the NHS don’t make any offers and your case proceeds to trial, then there are two things to consider:
A) how quickly you issue your claim in court. You have to issue within 3 years, but you may well issue sooner. It’s as soon as you’re ready.
B) once in court, depending on value and which court you issue in, you’re probably looking at 12-18 months before a trial. Could be sooner if it’s a simpler case, but I’m afraid it is not a quick process.
You’ve every right to bring a claim, and while it is tax payers money, I think it comes out of a different pot of money to nhs clinical funds.