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Have you tried using Disk Utility> First Aid on your Disk.

Boot to your Recovery HD (restart while pressing and holding down the Command and R keys).

When ready click on Disk Utility click on View in the menubar and select Show All Devices.

In the left hand panel click on the Disk (Apple HDD... or similar) and click on First Aid.

Then highlight on Macintosh HD and click on First Aid.

Run First Aid until you get the green tick for all Disks and Volumes.

Then try restarting normally again.

A bootable USB installer is a very handy thing to have in case of emergencies like this.

Everyone should have one.

To create a bootable USB installer you would download the OS your mac is running

macOS High Sierra for example by following the link here,

How to get old versions of macOS - Apple Support

Once you have downloaded the OS you want to install you can then make a bootable USB installer

by following the instructions here,

How to create a bootable installer for macOS – Apple Support (UK)

When you boot to the USB you can then use Disk Utility to run First Aid,

or Install the OS or Restore from a Time Machine backup.

When you get your mac up and running I suggest you create a bootable installer

so you can be in charge of any reinstallation's rather than rely on Apples Recovery Servers

which can be unreliable and very slow as you are finding out.

Where are you getting your information regards Internet Recovery.

Are you posting your question on a mac?

Sounds pretty catastrophic if it will not boot to the Recovery HD.

Don't suppose you have a bootable USB installer at your disposal.

If so you could boot to the USB and use Disk Utility from there.

Do you have any other macs in the house.

If you go ahead with reinstalling the OS through Internet Recovery you will lose all your data that is on your Disk.

This should be a last resort only after you have exhausted all other options.

If it is saying about 24hours then your mac must be using WiFi to do the reinstall, this is not a good idea.

If you want to reinstall the OS you really need the mac to be connected to your router via cable.

You haven't answered my two questions in my second post.

If you were able to boot to a USB installer and run Disk Utility> First Aid there

is a slim chance that it may be able to repair the Disk directory and bring back functionality

to your mac.

Prior to this failure was your mac running okay. If it was getting slow or being 'grumpy' then that

may have been an indication that your macs HD was failing.

Each time you upgrade your macs OS it is a good idea to create a bootable USB for the new OS.

A 2011 iMac can only upgrade as far as macOS High Sierra.

As you have another MacBook I would follow my instructions to download the OS that your mac was using earlier

create a bootable USB installer as described in the article linked to earlier.

Use that to run First Aid from Disk Utility to see if that can resurrect your mac, maybe too late as you

have started this Internet Recovery nonsense.

My only concern is newer macs running OS's like Big Sur or Monterey no longer allow you to download older OS installers. (Apple being their usual helpful selves). But don't let that stop you from trying

I may be wrong.

I have never had to use Internet Recovery because I make bootable USB installers.

If you still want to go ahead with Internet Recovery then I would stop it now

connect you mac to your router via ethernet cable and try again, the length of time

should be greatly reduced.

If WiFi is the only option then have your iMac as close to the router as possible

and if at all possible try not to use any other devices that share the same internet connection.

If you can't do that then try and limit the use of other devices to light duties.

No big downloads or streaming movies or TV.