Just make sure you are booting to a macOS 10.13+ installer since only macOS 10.13+ has the necessary NVMe SSD driver needed to work with the OWC NVMe SSD. When booting into Recovery Mode or Internet Recovery Mode, some Macs may only boot to the online installer for the OS which originally shipped with the Mac from the factory which for a Late-2013 Mac would be macOS 10.9 Mavericks. If you are booting a macOS 10.13+ installer, then make sure Disk Utility is showing the physical drives. Within Disk Utility you may need to click "View" and select "Show All Devices" so the physical drives appear on the left pane of Disk Utility. If the physical SSD does not appear in Disk Utility, then you have some sort of hardware issue (most likely with the SSD itself and assuming you are running macOS 10.13+ to check).
The original Apple SSD in this Mac was a SATA based SSD which may explain why it can be seen, but the OWC NVMe based SSD is not seen.
I do know that an early version of the OWC Aura SSD requires an SSD firmware update to work with later versions of macOS. This SSD firmware update is hidden away within an OWC knowledgebase article ....not sure why it is not on the official download site except it has a Beta designation:
https://eshop.macsales.com/Service/Knowledgebase/Article/10/730/Aura-SSDs-Firmware-Update
Unfortunately there are no adapters or enclosures that can be used with the OWC NVMe SSD. OWC has a warning on their Envoy Pro Enclosures that using the OWC Aura NVMe SSD in the Envoy Pro Enclosure will likely damage both the SSD & the enclosure. The only way to directly access data on the OWC NVMe SSD is by installing it internally in a Mac. Otherwise you will need to contact a professional data recovery service such as Drive Savers if you don't have a good backup.
People should always have frequent and regular backups of their computer and all external media (including the cloud) which contains important & unique data. FYI, it is impossible to recover accidentally deleted data from an SSD after the Trash has been emptied. Plus an SSD can fail at any time without any warning signs (even a brand new SSD).