I have the opposite problem. I have been receiving my UK state pension for more than 1 year (my total income is quite a bit higher than the tax allowance) but my income tax has not increased.
I informed HMRC online, who confirmed that I was not paying sufficient income tax, but they could not adjust my tax code without confirmation from DWP. I was told by HMRC that I need to request DWP to inform HMRC of my state pension income. I then contacted DWP but was told that HMRC already have access to my state pension information, and that I should contact HMRC myself. I then logged into my Government Gateway account and informed HMRC of my additional income.
This was 6 weeks ago and I have not received a response. Meantime, I am expecting a tax bill of 2K GPB or more when they eventually get around to sorting things out.
Lucky for me that I noticed the underpayment and set aside some money to pay the shortfall.
But, what an absolute shambles!