You won't get the Form 1099-R until January 2024. You should file your 2022 tax return before then, but I would at least request a filing extension so that you can file after the amount distributed to you is known.
"According to them, they will issue a check and that I have 60 days to redeposit it back into the Roth IRA."
This is wrong information. The return of excess contribution itself is not eligible for rollover. You will instead be rolling over the original distribution from the Roth 401(k). However, the normal 60-day deadline to roll over the distribution from the Roth 401(k) has long since passed, so your rollover needs to be done by the financial institution accepting self-certification under IRS Rev. Proc. 2020-46 that the rollover of the amount originally distributed from the Roth 401(k) (not the adjusted amount distributed to you from the traditional IRA as a return of excess contribution) would qualify for a waiver of the 60-day deadline as measured from the date of the August 2022 distribution from the Roth 401(k). I described that above.
