As several have noted, a card reader may be a cheap fix. If you buy (or borrow) one, its use may help diagnose the issue. If the card reader can see the photos, then the cord may be the culprit. If the card reader can't see the photos, then maybe the camera isn't writing the files properly - it could be the camera or the card. Another diagnostic test is simply take a few photos with another card in the camera and see if you can see them. If you have Lightroom or Elements, can it see the photos on the card? If not, how about a free program like Faststone?

I know you want to retrieve the photos from the card, but how long has it been since you reformatted the card?

We have ordered a card reader for future use. I did try to upload the photos from the camera to Lightroom and it didn’t recognize a camera either - still on the same computer. But the camera is recognized on my laptop, a different computer.
As for reformatting our card, we do it every time we delete photos from the camera after every use.

The card is a fairly new 128 card - which has worked previously.

I will be going through other suggestions shortly.
I really appreciate all of the suggestions to correct the problem.

Couple of areas to check. Check the "task manager" so see if Canon utility is loading correctly. Then locate "services" as you'll need to scroll down the list and locate any reference to "device manager" "device setup" "device Install services".

This is a known issue with WIN10 as the update tends to shut all these types of connective services like Wifi, wireless, bluetooth and the device settings.

You'll need to check or uncheck these device settings with a drop down, automatic, manual or disable etc. try each setting when connecting your camera. Also, you may need to connect via a designated USB port located at the rear of the computer. As there maybe 2 different USB versions located on your system.

I have a Canon 5D Mark III camera that I have had no problems in the past connecting to my Dell computer and downloading photos. Then all of a sudden a week ago - the computer quit recognizing the camera was attached. My EOS Utility program that downloads my photos from my camera to my computer no longer recognized my camera was connected and in my computer settings my camera was no longer recognized as being connected. I have uninstalled my EOS Utility several times - hoping this was the problem - but to no avail. I even downloaded the program from the Canon website - but it still doesn't work. Thinking it might be a driver, (I checked my shutter count on my camera with a site someone had suggested on this forum - and had to change a driver), I reinstalled a driver for the EOS Utility program from the Canon website - but that didn't work either. If anyone has any ideas or suggestions to try - please help. (And no comments about it being a Dell and not Apple either, please:-) Thank you for any suggestions. Oh yes, my camera is able to download photos using my laptop with the EOS Utility - so I don't think it is my camera??? I have also changed connecting cords and that doesn't help either.

Since you can download to your laptop, it cannot be a problem with your camera, card, or cable. Sounds like a glitch in the EOS Utility program, but I wonder why you even need such a program just to download? You might try connecting your camera to the computer, open your hard drive window (My PC, This PC, or whatever silly name they call it), turn on the camera and a folder should come up. If so, open it up to get to the images, click and drag them to a folder on your hard drive, after which you can open them with whatever post processing program you have. That’s how I do my digitals, hope it works for you.

*facePalm* I totally overlooked one key point that should have been mentioned. If the photos are in RAW format, the computer may not see them as I've encountered this after the purchase of the new camera. simply because I didn't have the correct software loaded. In other words, the image is on the SD card in a RAW format but without a reader or soft to read it, it will not show the image. I'm not sure this is your issues as you've been saying the camera is not being recognized when connected. But when you said that you went to your task manager and it shows that the device randomly shuts down or reverts back to a previous setting, has something to do with an on going issues with Win10. The best you can hope for is to extract the SD card and use a card reader or some computers even have a SD slot located on the front of the computer. Either way I've encountered these same issues when I updated to win10 and my issues was corrected when I readjusted the devices in the task manger.