Whether your notice pay has to be taxed depends on the terms of your contract and your employer’s custom and practice. If your contract contains a pay in lieu of notice clause (PILON) clause permitting your employer to pay your notice instead of making you work it, then the payment of notice will be taxable. This is irrespective of whether you have scope in your £30,000 tax free allowance to receive it tax free.
If there is no such clause, then it is usually paid tax free as your employer is in breach of contract by making the payment which most people are happy to receive rather than work anyway. The only exceptions to this are where the employer does not have such a clause but pays people their notice as routine which means it has become an implied term. If the amount that you will receive under the settlement tax free has already used up your £30,000 tax free allowance, then irrespective of your contract it will have to be taxed.
Your solicitor should be able to provide some advice on whether the payment should be taxed or not and would normally need to see your contract of employment as a starting point unless you are so far over the tax-free allowance that this would not make a difference.