The insurer should provide you proof of NCB - you pay for a policy and a NCB directly affects your premium - so if you have earned it, they should give it to you so that your new premium can be adjusted to take in to account the NCB discount.
Given the dispute with the insurer, you can avail of the Financial Ombudsman Service dispute resolution scheme - which is much quicker than court action and free to use.
You have to go through a complaints procedure first. If the complaint resolves this, you can show your new insurer the proof of NCB which should result in a rebate of any further money you had to pay (the premium would be higher due to no proof of the NCB).
You are covered under the Consumer Rights Act 2015 here which means you have a right to expect a service (from the insurer) carried out with reasonable care and skill. Furthermore, the policy should be “fit for purpose” and “as described”. If it isn’t then you can allege breach of your consumer rights and breach of contract.
You should now make a formal complaint and if that does not resolve the matter ask them for a "deadlock letter", which is a letter giving their final response. Details of how to make a complaint will be on their website if you take a look.
When you do make the complaint, tell the insurer you plan to report them to the Financial Conduct Authority (FCA) who regulates them. You can reach the FCA to report them on 0800(###) ###-####if you wish. It may speed matters up.
Once you have the deadlock letter the next step would be to escalate this to the Ombudsman - they will investigate and liaise with the insurer which would hopefully result in a resolution - they should provide you with the proof of NCB. They can order the insurer to make a financial award for inconvenience if they have acted poorly.
Once you have their final response, you can make the complaint to the Ombudsman here : www.financial-ombudsman.org.uk/consumers/how-to-complain
Or by email:***@******.***
Based on the circumstances, assuming the insurer does not uphold your complaint, I am sure the Ombudsman will do. The Ombudsman will look at this case independently and will make a decision based on what happened. Their decision is binding on the insurer.
If the Ombudsman did not find in your favour, therefore, you then have the option of suing the insurer for breach of your consumer rights and you have up to 6 years to bring a claim to the court (the limitation period is 6 years from your date of loss, to when you need to issue a claim). You could claim the sum which you have had to pay due to the NCB not being taken in to account with your new premium.
Court action is considered as a last resort though - I am sure the Ombudsman will resolve this given the facts.