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.
Make sure the mac is connected to your router via a cabled connection, WiFi is too unreliable and will probably reset the date back to today.
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 011421002016
press Return
enter your password
press Return
If Terminal returns an error saying sudo : command not found, then try again without sudo.
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.
The alternative is to borrow a mac that is running El Capitan or at one time could have ran El Capitan.
On the donor mac, click on the link below.
How to upgrade to OS X El Capitan – Apple Support
Go to Section 4 and click on Download OS X El Capitan.
This downloads InstallMacOSX.dmg to your Downloads folder.
When downloaded open to InstallMacOSX.pkg, double-click on
that and an installation window will open, this does not install El Capitan
but converts the .pkg to the Install OS X El Capitan.app which you will find in
your Applications folder, it should be 6.2GBs in size.
Now that you have the Install OS X El Capitan.app in the Applications folder you can make
a bootable USB installer disk, read the instructions here,
How to create a bootable installer for macOS – Apple Support
The USB should be 8GBs and called MyVolume.
Once you have created the bootable USB disk then you can plug it in to your wife mac.
Restart the mac while pressing and holding the option/ alt key, this should
open the Startup Manager which will show you which disks are bootable.
Select the USB, it should now be called Install OS X El Capitan and be orange.
Press Return.
The mac will now boot to the installer USB, a Utilities panel will open
select Install OS X and press Continue.
The installer screens will open and you will be guided through, follow the prompts.
