upgrade OS of mac mini late 2012

I'm afraid you've landed in a Catch-22.

According to https://support.apple.com/HT211683 you have to first install macOS Mojave 10.14. Then you can install macOS Catalina 10.15

However these links are https. (secure) Safari on your Mac cannot access https sites because the security certificate has expired. Firefox can access https sites as it uses its own certificate, but it cannot open these links (which need to open the Mac App Store). So basically you cannot install the upgrade until you have installed the upgrade.

However, according again to that help page, you can download macOS Sierra 10.12 installer, which comes as a direct link to a disk image and which, oddly, is http not https. If you install this you may be able to access the Catalina installer, but I have a nasty feeling that you might still be stuck with the expired certificate and be back where you started: probably worth a try. (I'm not sure, but possibly Sierra is the oldest system not to have an expired certificate.)

There is a workaround for the certificate, but I emphasize that I haven't tried it and don't know whether it's effective or even advisable:

• Download the ISRG Root X1 certificate file from http://x1.i.lencr.org/
• Open the Keychain Access app and drag that file into the System folder of that app.
• Find the ISRG Root X1 certificate in System and double click on it, open the Trust menu and change "Use System Defaults" to "Always Trust", then close that and enter your password to confirm the change (if prompted).

Let's Encrypt DST Root CA X3 expiry Sept 30th 2021 | Certify The Web Docs

In the second line they presumably mean the 'System'>'Certificates' section of Keychain access.

If you don't feel like performing this fandango you could try Apple Support. I don't know whether they would be willing to provide you with a download of the required installers, but asking may be your only option, unless you can get someone with a more up to date Mac to download and save the installers (which if not stopped would install and then self-erase). An Apple Store might help. Either way you would need a suitably sized USB memory stick (around 5GB per installer).

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Might suggest moving to at least High Sierra. It should better position the machine with Boot ROM updates needed for the drive conversion to newer APFS Format. The format will be need if moving to Catalina.

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina & Big Sur supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal operation of the OS and may even Corrupt the OS requiring a Reinstallation

Everymac.com says the OP's computer can be upgraded to Catalina. However his Safari won't access the installers required including High Sierra as the URLs are https and the security certificate has expired. Firefox won't access the App Store from the links so I assume other browsers won't either. He may be able to download Sierra, and that may have a valid certificate: if so from there he would be able to access the Catalina installer.

Connect with an Ethernet cable.

Startup in Internet Recovery Mode.

Open Disk Utility and erase the Mac HD.

Quit Disk Utility and go to Install macOS.

At that point you should be offered two options.

Select the “latest macOS” option and install that.

Excellent point - as usual :-)

The OP best follow your advise

I don't know the answer to that one, for certain but as it's a dmg and not a link to the App Store it ought to work in theory.