Old lightroom and new camera

Borrow a friend's (recent OS) computer, install the Adobe DNG Converter, convert your Z6 files, copy the DNGs back to your system.

Oh and another problem is that even if you are able to update Mac OS X, you might not get the very old Lightroom 5 to run on it so it's a real conundrum.

You need at least Mac OS X 10.14 for Studio NX. I don't think there is any program from Adobe or Nikon that can read Z6 files that runs on Mac OS X 10.9. I think if you are not able to upgrade the operating system or computer, you are stuck. You might be able to get something like gimp image editing (open source) running and that should be able to open Z6 files. Lastly, 10.9 is no longer supported with security updates by Apple so if you are connected to the internet with that Mac you are not safe from being hacked as I think there are known exploits of old versions of Mac OS X that are only patched in newer versions. So be careful.

I think it should have been included with the camera, but if not it's called NX Studio. It is free software, and can be downloaded from the following link:

https://www.nikonusa.com/en/Nikon-Products/Imaging-Software/nx-studio.page 

Only problem is, it seems that this new version doesn't allow you to save the image as a TIF. I just downloaded it myself and haven't had an opportunity to experiment with it. But at least you can edit your NEF files.

Update: it's necessary to use the Export command to create the TIF file. It works.

I'm a Windows user, so I'm unfamiliar with the Mac OS. If you are unable to upgrade your OS then it seems that your choices are limited. You can shoot JPEG images with your new Nikon and process them in Lightroom 5. You can use the Nikon software to process the NEF files and create TIF files that can then be opened in Lightroom 5. Or, you can purchase a new iMac. There are no updates for Lightroom 5 that will enable you to work with the NEF files from that camera model.