HP 5.1.1 Printer Software Update - Apple Support
Earlier reports indicate that the use of an HP LaserJet 1022 driver from that package is possible with an HP LaserJet 1020 printer as well. However, the text says Not compatible with macOS v12 and newer, so do not expect it to work with macOS Sequoia 15 without modifications. If the package will not install, various solutions have been suggested (try a community search, for example, for the HP LaserJet P1005), but the success rate is somewhat unclear.
If a compatible HP driver for macOS 15 cannot be found, or if another software solution is not possible, you may have to look for a workaround.
Linux drivers are available for the HP LaserJet 1020.
https://developers.hp.com/hp-linux-imaging-and-printing/supported_devices/index
If nothing else helps, you could perhaps investigate whether a small Linux-based Raspberry Pi print server would allow a USB-connected HP LaserJet 1020 to be shared with a modern Mac (or iPhone/iPad) in an AirPrint-like way. Look upon this as an experiment.
Tesserax reported about using a Raspberry Pi 3 Model B+ for an old HP LaserJet printer.
Can I connect my laser printer to my netw… - Apple Community
Details about a Raspberry Pi print server setup (for a USB connection to a Kyocera laser printer) can be found in the posts by Techguyuk in Can’t print with iPadOS17 - Apple Community.