Bootcamp Mojave and Windows 10 now all Mac OS startup keys don't work
It sounds like you updated the firmware on the video card. The drive selection screen and other startup keystrokes work from the original EFI. This likely happened when updating W10. You may have accidentally also updated a video driver that included a VC firmware update. Been there, done that. Getting the card re-flashed to the Apple VC EFI is likely to be difficult and/or expensive. This is especially true since this is an iMac. The video card is integrated into the logic board. My solution won't work for you, unfortunately.

Perhaps an Authorized Service Provider can flash it. If you're not sure where to go, try Apple first. Call (800-275-2273) or go to the support site and when you get through all the repair prompts, it will show a map of the services near you. Perhaps the Genius Bar can flash it, too. Don't let 2013 deter you from calling. They may be able to help on the phone (doubt it), but they can at least help you find a place to get it done.

If you have not been able to start the Mac OS, as a Hail Mary, try holding 'x' during startup to see if you can get the MacOS to start (which you may have already). Or use Target Disk Mode to remove the Windows partition. When the Mac can't find an OS on the selected startup disk, it looks for a new one and it may startup from the Mac OS. Then you can use Sys Prefs to change your startup disk back to the Mac OS.

However, if you no longer have a Mac OS (deleted partition maybe?), then it will start in Internet Recovery. If you have a Time Machine backup, you can restore the your original Mac OS and content using your backup. Otherwise, you will have to reinstall the OS.

Let me say this. My experience with the issue (update Windows, flash card, no startup keystrokes) does not fit your scenario exactly so I could be totally wrong. If you don't have a backup and have data to retrieve, try Target Disk Mode before making any changes. You can save important data (assuming it still on the hard drive) to an external hard drive.

Good luck.