Imac not responsive after Internet recovery

Agree with @BDAqua that it may take a while to boot into Internet Recovery Mode depending on the speed of your network & Apple's servers at the time. Plus, if the internal hard drive is failing, then it could be causing a severe slowdown.

You can try running the Apple Diagnostics to see if it boots and whether any hardware issues are detected.

It would also be nice to know the exact model of this Mac and version of macOS installed. You can get the exact model by entering the system serial number on the check coverage page (please do not post the serial number since it is considered personal information):

Check Your Service and Support Coverage - Apple Support

Start up from macOS Recovery

Command (⌘)-R

Install the latest macOS that was installed on your Mac.

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

Look at top of screen for Utilities menu...

In Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

Reinstall the OS.

Once you boot into Restore Mode...

Look at top of screen for Utilities menu...

In Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

Oh, just to mention lately MacOS installs can look like they're staled & doing nothin, but usually contnues, then also, besides the Mac testing your patience, it may restart up to 7 times & testing your patience some more with things like 39 minutes remaining the 50 minutes later restarts & say 29 minutes remaining.. then... less tan a minute remaining & an an hour later the minute gets used up & you see a Desktop finally!:)

Did you run Disk Utility?

Did you use... Option+⌘-+R ?

Thanks.:)

How about the Disk Repair?