iOS 12.5.5 software show but not update automatically

Evidently, as your iPad Air is only offering you 12.5.5 (the latest version of iOS (now called iPadOS for iPads) is 15.4.1). This means that you have the original (1st generation) iPad Air (the iPad Air 2, and later, can be updated to 15.4.1). 12.5.5 is a legacy version of iOS/iPadOS which contains bug fixes for devices unable to run later versions. Apple has stated that 12.5.5 will be the last version of iOS released for these devices. Your iPad, like the iPhone 6 (and older models), only has 1GB RAM, all recent apps will require a device with 2GB RAM. If you want to download new apps (or update your current ones if the developers are still supporting them), you will need an iPad with 2GB RAM.

My recommendation is that, as your iPad Air is now considered antiquated (it was initially released in November 2013) you should replace it. That said, what happens when you try to download iOS 12.5.5 manually? Have you tried installing it by connecting your iPad to your computer and using iTunes?

Some things you can try. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch    *

Try a forced restart. See the section towards the end of the article.

Reset Device. -