Remove the old SSD from your Mac.
Insert the bootable USB with Mac OS 10.13 on it and boot into recovery mode.
Go to Disk utility.
Connect the external HDD (make sure it appears in disk utility).
Install Mac OS 10.13 from the bootable USB stick onto the external HDD.
After installation, turn the computer off.
Disconnect bootable USB stick and external HDD and install your new NVME SSD.
Connect the external HDD and start up your Mac (your computer will use the external HDD with its newly installed Mac OS 10.13 to boot up).
Go to Disk utility and from the left hand side, upper corner, collapse the drop down menu and choose 'Show all devices'. Your new NVME SSD should be visible now.
You can now carbon copy Mac OS 10.13 from the external HDD onto your new SSD using Disk utility or some other 3rd party software.
Once finished, turn off your Mac. Disconnect the external HDD and start up again. If all has gone well, your computer should boot from your new SSD with Mac OS 10.13. You can then clean install mac OS 10.13 if you wish by downloading it from Apple website, or upgrade to the newest version of Mac OS.
* The host computer itself must be upgraded to 10.13 before the new NVME drive is installed. This operating system contains an EFI firmware update which adds host support for these drives. Without this the Mac won't know what the SSD is.
**AKA High Sierra