Sorry to hear of the issue. I presume the insurer is refusing to provide cover given the car tax issue.
The policy terms and conditions are likely to say that cover is contingent on the car being roadworthy and taxed. So there could be an issue if the evidence is the car was untaxed on 4th November. You say that the record shows the car insured from 1st November which is fine - the issue is the lack of tax.
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.
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 compact will be on your insurer's 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 reinstatement of cover under the policy. 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:***@******.***
The Ombudsman will look at this case independently and will make a decision based on what happened. Their decision is binding on the insurer.
This process would not require a lawyer either. If the Ombudsman did not see it your way, you could consider court action though a full assessment of the facts would need to be made by a lawyer ideally, before the court is involved.