This is my take on the situation:
Absent the new plan being able to correct this as a bookkeeping error, the amount from the old Roth 401(k) account deposited into the new traditional 401(k) account must be treated as a excess contribution to the new traditional 401(k).
An In-plan Roth Rollover can't be used to fix this because an excess contribution is ineligible for rollover.
Removal of the excess contribution must include a pro rata share of investment gains. Unfortunately, the pro rata calculation must be done over the entire 401(k) balance, so having kept in cash the portion that came from the Roth 401(k) does not simplify things. Any investment gain that accompanies the distribution of the excess will be taxable in the year distributed and cannot be rolled over to any retirement account.
Because the funds from the old Roth 401(k) account were not deposited directly into the new Roth 401(k) account, the distribution of the funds from the old Roth 401(k) account failed constitute a direct rollover. Because moving funds from one Roth 401(k) account to a Roth 401(k) account in a different plan is only permitted to be done by direct rollover, it won't be possible to do a late rollover of these funds to the Roth account in the new employer's plan. However, an indirect rollover to a Roth IRA is permitted, so it seems that the only way to get the funds back into a Roth account would be to find a Roth IRA custodian that will accept self-certification under IRS Rev. Proc. 2020-46 that this late rollover would qualify for a waiver of the 60-day rollover deadline due to one of the reasons listed in the Rev. Proc. and roll it indirectly to a Roth IRA.
https://www.irs.gov/pub/irs-drop/rp-20-46.pdf
The old employer erred in not issuing a separate check for each portion of the rollover and treating these as independent rollover requests. Although it's often done improperly, a proper direct rollover specifies the account that is to receive the direct rollover, and it's impractical to specify multiple destination accounts as being payees of a single check. In the case of a direct rollover of a Roth 401(k) account, the rollover information also needs to specify the amount that is contribution basis.
You should be receiving from the old plan a code-G Form 1099-R for the portion distributed from the traditional account and a code-BG Form 1099-R for the portion that was intended to be a direct rollover from the Roth account. Technically you should submit a substitute Form 1099-R for the code-BG Form 1099-R since this failed to be a direct rollover. This substitute Form 1099-R will need to have code B (along with code 1 if you were under age 59½). Assuming that you are able to complete a late indirect rollover of this distribution to a Roth IRA, you'll report this distribution as having been rolled over. However, assuming that the rollover to a Roth IRA is completed, simply reporting the code-BG From 1099-R as received will produce the same taxable result, but TurboTax's tracking of your basis in Roth IRAs will be incorrect so you would need to correct that.
