New iMac Bluetooth mouse not connecting

see > If your Apple wireless mouse, keyboard, or trackpad isn't working as expected - Apple Support

FWIW I always keep around an inexpensive Logitech USB Keyboard and Mouse for troubleshooting the fancy branded ones.

Sorry to hear about your loss. Dealing with a computer issue is probably the last thing you want to be doing. Since this is a relatively new iMac I suggest calling Apple directly so you can chat with a support person who has probably dealt with this type of issue more times than any of us on this discussion board.

Contact Apple for support and service - Apple Support

Before you call them there are a couple items to try. Make sure that after powering on the iMac you are clicking on the mouse to wake it up. Try a minimum or two or more clicks.

Turn the mouse over and you'll see a small ~1/4 inch hole with a clear lens in it just to the left of the power switch. Make sure there aren't any fibers, pieces of dust, or a trapped piece of hair on or near the lens or it could prevent the mouse cursor from moving. It's likely not the issue, but it is easy to check just in case. In my case I once had a piece of hair on the lens which prevented the mouse cursor from moving side to side. Removing the tiny hair resolved the issue immediately. If you are able to locate your old mouse or borrow one from a friend that will quickly determine whether the problem is the result of a mouse issue or an issue with the iMac.

Try resetting the NVRAM which is a small amount of non-volatile memory that stores some variables. Resetting the NVRAM has been known to resolve issues and is easily done by powering on the iMac and then pressing these four keys together: Option, Command, P, and R for ~20 seconds. More info can be found in this Apple article:

https://support.apple.com/en-us/HT204063

Good luck, and please let us know if you need additional assistance, and especially when you get your iMac back up and running.

Definitely follow the steps in the article that den.thed suggested.

You mentioned that you charged the mouse two different ways. Did either of those two ways involve connecting the mouse to one of the USB ports on the back of your iMac? If not, turn the mouse on and connect it to the back of the iMac using the USB to Lightning cable. Not only does this charge the mouse, but it will also pair and connect it automatically if for some reason the pairing/connection was disabled.

With the mouse connected to your iMac turn the mouse power switch off. Does a dialog pop up telling you the mouse is off? If it does then this indicates that the mouse is powering on/off properly and the iMac sees it. Leave the mouse powered on and disconnect the Lightning cable to and check whether the mouse begins working properly.

Are you using an Apple Wireless keyboard? Is it working properly? If the keyboard is connected to your iMac using a USB-Lightning cable then try disconnecting the cable to verify that it will also work wirelessly. If it stops working then that is an indication you might be having problems with Bluetooth in your iMac rather than a problem with your mouse. The article den.thed has a table showing the various Bluetooth icons you might encounter. What does your Bluetooth icon look like? It may provide clues about the issue you're trying to resolve.

Click on the Apple Support article in blue lettering that I posted above and see if anything in the first section helps you get your keyboard and mouse working.

If not, then you actually may want to dig out your USB mouse or buy an inexpensive USB Logitech keyboard and mouse combo from your local Best Buy, Staples, Target or Walmart, etc. see > https://www.logitech.com/en-us/product/desktop-mk120

Which is a whole lot easier, than packing up the iMac and driving to the Apple Store.