External hard drive not mounting on new iMac

Do you have Finder configured to display mounted volumes on the Finder's Sidebar? You can also select "Computer" from the Finder's "Go" menu which should also show you any mounted volumes.

Is the external drive even mounted? Check Disk Utility to see if the external drive is mounted or even physically seen. Within Disk Utility you may need to click "View" and select "Show All Devices" before the physical drives appear on the left pane of Disk Utility. Sometimes an external drive may need to be manually mounted using Disk Utility (especially if the drive is encrypted).

Is the external drive connected directly to the new Mac?

Does the external drive have its own dedicated power supply?

Try disconnecting all other external devices in case one of them is causing a problem.

Try booting into Safe Mode to see if you can access the external drive. If the external drive is using NTFS and you have any third party NTFS drivers installed, then you may not be able to access the NTFS volume while booted into Safe Mode since those third party drivers may modify macOS settings to only allow the use of the third party drive which would be disabled in Safe Mode.

Maybe the USB adapter or cable being used with the external drive has an issue.

Perhaps the external drive is not receiving enough power and either requires its own power adapter, or perhaps a powered USB3 hub can be used to give the external drive more power.

Is the new iMac an Intel or M-series Mac?

What is the exact make & model of this external drive?

And what file system is on the external drive?

Were you using any third party software to manage this external drive on the broken iMac?

I think the age of some of these external hard drives (a number of similar anomalies have been reported in the Discussions) may be a part of the problem. I have never seen this issue with any of the 3 Intel MacBook Pros and 1 M3 MacBook Pro with which we use numerous external bus powered drives ... until I just saw it with the new M3 MacBook Pro and an ancient Lacie Drive which is not bus powered but has its own power supply. That Lacie drive does not appear anywhere in that M3's Disk Utility or anywhere, although multiple other drives do appear. The Lacie is so old it still uses one of these 5 Gbps connectors

I think it is easily more than 10 years old, maybe more.

Connecting to the MacBook Pro M3 through an Apple USB-C to USB 3 adaptor (this adaptor works fine with multiple other external drives so I think the adaptor is fine), it won't appear or mount - recall it has its own power supply. But connecting through an OWC dock via one of its USB 3 ports, it mounts immediately on the M3 MacBook Pro.

The drive has been used for Time Machine for years, it is APFS as set up by Sonoma. It has no problem connecting and mounting on one of the MacBook Pro Intel Macs through that same Apple USB-C to USB-3 adaptor.

So I don't trust it anymore but will attribute this to drive degradation/malfunction, aging, etc. it has been running continuously for 10 hours/day for more than 10 years. Still, it is odd that it won't connect to the new MacBook Pro M3 but will mount on the 2019 MacBook Pro Intel ... with the same USB-C adaptor. But it can connect to the M3 MacBook Pro through an OWC dock. Does any of this make sense to you?