Sorry, you may have been provided with incorrect information depending on your circumstances.
From the wording of your query, we are assuming you are already in receipt of your state pension. No matter how long it takes HMRC to allocate the funds to your National Insurance record, HMRC will advise the pension service of the date we received the payment. This ensures that your pension is backdated to the date the payment was received.
If, however, you have not yet started to claim your state pension, the payment will only increase your pension from the date you reach state pension age.
If you are concerned your account may not have been updated correctly and your pension fully backdated, we would advise you to contact the National Insurance Helpline.
National Insurance: general enquiries