Mac OS X Lion not installing

The certificates for several of Apples OS's expired in October 2019, Apple have been very poor at getting these updated on their Recovery Servers with valid certificates. This may be the problem you are experiencing. 

Try this workaround, we are going to set the time and date on your mac to a date previous to the expiry date of the certificates.

Boot to your Recovery HD, click on Utilities in the menubar select Terminal.

Enter a new date, for example or just copy and paste

sudo date -u 011421002018

press Return

enter your password

press Return

Now try downloading the OS.

If this works then when the OS is installed and booted up you can Open System Preferences> Date & Time

and reset the time back to today.

Let me know if you get any errors from Terminal.

Just an update in Terminal type

date -u 0114210018

press Return

Does the date change to January 14th 2018 21:00hrs.

Is your mac connected to your router via cable, do not rely on WiFi.

After connecting the ethernet cable and booting back to your Recovery HD, did you go back and reset the date as described previously. And then try downloading OS X Lion.