I'm not sure the retirement date is the best starting point, so I will give you the milestones. My benefit commencement date was 09/01/22. I completed the paperwork electronically and it was submitted on 08/24/22. I received a confirmation of my benefits choices on 08/30/22. My financial institution received the pension lump sum check on 10/18/22.
Also, you can receive different checks for taxable versus non-taxable (contributory funds). You shouldn't deposit the non-taxable in a traditional IRA because then it will be taxed again when withdrawn. In my case I did not, but because it was clearly documented, my financial institution was able to separate the money.