The weekly state pension is taxable. The amount of tax, if any, you will pay on your state pension lump sum will be based on your top rate of tax in the tax year that you take it. The calculation of your income for this purpose will include all sources of your income but excluding the lump sum itself. Take a look at the example under the heading "Tax years 2008/09 onwards" here.In practice, you add up all your income in the tax year you take the lump sum but excluding the lump sum itself and deduct your personal allowance which is currently £10,000 or £10,500 or £10,660 depending on your date of birth (see here).If your personal allowance exceeds your income, your top tax rate will be 0% and you will pay no tax on the deferred lump sum.If your income exceeds your personal allowance but your total income after the deduction of the PA is £31,865 or less (this changes from year to year) you will pay tax at 20% on your deferred lump sum.From what you have said, you probably won't be a 40% taxpayer. If you can wait until a tax year in which you are not working so that you only have your weekly state pension and your private pension, you may be able to avoid paying tax on your lump sum altogether.I hope this helps but let me know if you have any further questions.
As I said in my answer, the lump sum will not be income for the purpose of determining your top tax rate which could be 0%, 20%, 40% or 45% in any tax year that you take it unless the rules change which is unlikely.What will be income for the purpose of determining your top tax rate in the tax year that you take the lump sum will be the private pension and the weekly state pension that you will receive on top of your lump sum. Some people forget about the weekly state pension. Don't do that.