bluetooth problem

Borrow a USB keyboard and reset the NVRAM: Reset NVRAM or PRAM on your Mac - Apple Support. If it doesn't help the first time try again a couple of times.

Having a USB keyboard is a valuable tool as a Bluetooth keyboard doesn't always start in time to let you boot into Safe Mode, the Recover volume, etc.

Glad you've got it under hand.

Do you happen to have any other Bluetooth devices enabled and active in the near vicinity other than this mouse and trackpad? How about Wi-Fi? For example, do you have a wireless router nearby your iMac?

Either or both of these could cause interference with these Bluetooth devices and possibly the reason why they are not appearing in the Bluetooth menu on your iMac. Also are these actually Bluetooth devices or using a proprietary wireless connection ... something like Logitech's Unifying wireless?

You're very welcome! I'm glad you got it worked out with OT's suggestion.