Failed OSX update and how to fix it
You may need two USB drives at this point. Your goal is to make a bootable installer for Catalina.

https://support.apple.com/en-us/HT201372

Assuming you don’t have another Mac handy or can’t convince someone you trust to make a drive and deliver/ship it, from the one you can boot to recovery, open the terminal app in utilities and see if you can run the guide above.
On older OS and some firmware, you could also format a USB as FAT (MS-DOS) and just drop the installer app but I don’t think that works for Catalina and you need the script to prepare things.
Even if your Mac can run internet recovery, since your network couldn’t successfully download the installer app from the App Store, something isn’t reliable in the setup so that might be more hassle than help. Also, you won’t know if hardware is the issue since failing to install is generally one of three things. 1) bad network 2) not the correct steps to erase and install 3) hardware.