The October 15, 2019 deadline was to receive the corrective distribution, which you've already done. What you you have left to do is amend your 2018 tax return to include the earnings which are subject to 2018 income tax and, if no penalty exception applies, to a 10% early-distribution penalty, and also to remove the 6% excess contribution penalty if you had originally reported the excess Roth IRA contribution.
To amend using 2018 TurboTax you can either enter the code JP 2019 Form 1099-R as you expect to receive it in 2019 (total distributed in box 1 and just the earnings in box 2a) or, as your tax advisor suggested, wait until you actually receive this Form 1099-R near the end of January 2020. You should also go through the IRA contribution section, make sure that you have entered the original Roth contribution then indicate that you had the original contribution amount returned. The filed amendment will be the same either way. This Form 1099-R does not get attached to your amendment. Your explanation statement regarding the contribution and it's return, including the amount of attributable earnings, is all that is needed to accompany Form 1040X (along with payment of a balance due, if any).
