Here is HP Download page for your PC: HP Omni 27-1010eg Desktop PC Software and Driver Downloads | HP® Customer Support with Windows 7/8.1 AMD Graphics drivers.
According to CNET your All-In-One PC has a Radeon HD6550A GPU card installed: HP Pavilion 27-1010eg - all-in-one - Core i5 2390T 2.7 GHz - 6 GB - 2 TB - LED 27" Specs - CNET
To verify the above information, Download and run GPU-Z which should show the exact AMD GPU card you have installed. If it is the same as above I couldn't find any HD6550A which according to TechPowerUP is a Mobile GPU Chip: AMD Radeon HD 6550A Specs | TechPowerUp GPU Database .
See if this AMD driver for the HD6550M Mobile chip will be compatible from AMD download page: https://www.amd.com/en/support/graphics/amd-radeon-hd/amd-radeon-hd-6000m-series/amd-radeon-hd-6650m
You only have two last AMD Drivers to use since your HD6550M is not supported anymore by Driver updates by AMD.
Also you must have all of your Intel drivers updated to the latest versions especially the Intel Graphics and Chipset and BIOS.
Try using HP Support Assistant to update all your Windows 10 drivers since HP doesn't have any for your computer: HP PCs - HP Support Assistant 9.0 (Windows 10, 8, 7)
I have attached the guide for the above HP Support Assistant to this reply if you are interested in checking it out.
Thank you all very much! I will work through the information and report back. 
AMD has no drivers for mobile HD5570 GPU chip.
What does GPU-Z show when you run it.
Try installing your HP's Windows 7 AMD Graphic driver and see if it starts working normally again.
Try to use Windows 7 drivers if there are no Windows 10 available.
When you do install the AMD driver use this basic method to remove and install the driver to prevent conflicts:
Uninstall the current AMD Driver using DDU as per this method:
It could be due to a corrupted AMD Driver installation.
Download free program DDU (Display Driver Uninstaller). Run it in Safe mode with the internet disconnected.
You can also run it in Windows Desktop but then you would need to Reboot the computer for DDU to finish erasing all of the AMD Driver traces from your computer.
Once it finishes uninstalling the current AMD Driver in Safe mode and boots back to Windows Desktop, still with the Internet disconnected, delete the AMD Installation folder C:\AMD if it was created before.
Now install the full AMD Drive package you downloaded manually from AMD Download page. Not the express package. Run the package and if it installs correctly again delete C:\AMD folder and reconnect the internet.
This should prevent any conflict when installing the new version from the previous version.
By the way, C:\AMD is always created whenever you run the AMD Driver package
Here is what CPU-Z shows:
Unfortunately, this is the outcome even after doing my best to follow every step of your advice. May I have downloaded the wrong driver package, or is there another problem?
I also had the idea of locating the old graphics card driver on my external hard disk. Should I give that a try?
Run GPU-Z and click on the "Lookup" button..
Sorry, I thought the "G" was a typo.
GPU-Z can't find my graphics card either. The Lookup button just brings me to a website where I can specify the card I am looking for, which seems to defeat the purpose.
I will now try to manually install the AMD drivers from the HP support page and report back.
"I also had the idea of locating the old graphics card driver on my external hard disk. Should I give that a try?"
Any advice on this?
Nope, no matter which compatibility setting I use, the driver installer gives me this message. Some other drivers were installed, but none of the ones that might be the correct driver for the AMD card.
