Claiming pension tax relief for previous years

Hello, I'm looking for guidance regarding expected processing time for the pension tax relief claim for previous years. I've been contributing to my NEST pension account for the past few years, and recently found out that they only claim back 20%. Higher rate tax payers need to claim the rest via self assessment, which is what I'm planning to do for the 2023-2024 FY year, as I haven't submitted my self assessment yet. However, claiming it back for the years before the previous one should be done by contacting HMRC directly. I've collected the info on my gross income and pension contributions for the past years and included this in a letter which I sent to HMRC on April 5 asking to confirm the receipt, but I haven't heard anything since. Can you please advise on the processing time for such requests and if I should be contacting HMRC via phone instead. Thank you in advance!

Hi multipassport23,
We do have a backlog at present and you would be looking at October for a reply.
Thank you.