Do you have a bootable backup? If so, you should try booting from that, and restoring your disk back to the state it was in before the attempted install. If it was me I would restore back to Mojave, and just run with that for a while until Apple works out the bugs in Catalina; you’re not the only person I’ve heard of that has had a completely failed install of Catalina.
If that is not an option you could also try going forward with Internet Recovery, as detailed in this Apple Support article:
Learn what to do if you can't get your Mac to start up from macOS Recovery.
support.apple.com
See the second section where it tells you what your options are if Recovery Mode has failed (as it apparently has for you).
Worst case would be no backup, and Internet Recovery Mode won’t work; in that scenario I would enlist the help of another Mac-owning friend. Have that friend create a bootable disk for you, and then focus your efforts on getting a copy of your data onto a second external drive (you can use a utility like SuperDuper to clone the internal disk to that second drive). Once you have a backup, even a backup of the failed install drive, then you can try moving forward with the internal disk, either to reinstall Mojave (recommended) or Catalina. If the installation works there will be a point in the process where Migration Assistant will offer to transfer data from an external disk, and that’s when you plug in your backup and tell it to use that.