Post P45 payments are taxed using a 0T Month 1 tax code. This means that, depending on the size of the payment, some of it may be taxed at 40% or 45% even if the taxpayer pays tax at 20%. This ensures that HMRC won't be left short as they were under the old rules for such payments and it also means that the taxpayer has to claim any excess tax back. You need to get in touch with your former employer to get the taxable redundancy gross and tax figures. HMRC may have them too as we are two months past the end of the 2015/16 tax year. Call them on the number here. Use boxes 5, 6 and 9 of the share schemes and employment lump sums section on page Ai2 of the SA101 pages here to report your redundancy payments if none of it was in your P45 or P60 figures.
As far as your current tax code is concerned, you lose £1 of your personal allowance for every £2 you earn over £100,000. If you earn £122,000 or more, you will have no personal allowance. If you earn between £100,000 and £122,000, you will be entitled to some of the allowance. Call the tax office to have your code reviewed and changed if necessary based on what you earn. An employer is not responsible for checking tax codes. That is the employee's responsibility.
I hope this helps but let me know if you have any further questions.