BACKUPS ON TIME MACHINE

You might be better off erasing the LaCie drive and then starting a new Time Machine backup. Otherwise you may end up with the old backups and the new backups. Which may or may not be a problem depending on how much storage is used. Eventually it will run out of space and start removing old backups but it won't touch the old backups from the previous Mac.

Alternatively, you can buy another LaCie drive and start fresh while keeping the older backups as a safety net.

To erase the LaCie open Disk Utility click the View menu and Show All Devices. Highlight the LaCie in the left-sidebar and click the Erase button. Format it with APFS as the new Time Machine uses APFS snapshots. If you format it with HFS+ it will work but not as efficiently as it will with APFS. Once erased, eject the drive, unplug it and plug it back in. Time Machine should prompt to use it as a Time Machine drive. Then just leave it attached overnight for the first initial long backup.

You should be able to use the LaCie TM drive with your new M1. As a test do the following;

1 - connect the LaCit directly to your new iMac via one of the ports. Make sure it shows on the Desktop.

2 - hold down the Option key. go to the TM menu and select "Browse Other Backup Disks…".

3 - go back in time to a file on the Desktop and see if you can restore it to the Desktop. If you have another identical file already select the option to keep both.

If this works then you can continue to use the LaCie with the new iMac. It it doesn't work your best bet is to follow James Brickley's recommendation. You can at any time in the future do what he recommends if you don't need any of the old backups.

Search Amazon for USB-C to USB-A adapter. Choose the best seller or Amazon's Choice. I see two packs that look like this:

Anker is a well known brand and I see blue on the USB-A side of the connector that means it's USB3 so it might be faster than the Chinese Best Seller.