davidneale1 wrote:
The Mac is an iMac 27 inch Mid 2010, 3.2 GHz Intel Core i3, with 12 GB 1333 MHz DDR3.
Try booting into Internet Recovery Mode using Command + Option + R to attempt to access the online High Sierra installer....assuming you had macOS 10.13 High Sierra installed on this system at some point in the past. You will need to use Fix #3 in the following article as well:
Unfortunately some Macs may only boot to the older online installer which originally shipped from the factory (or in this case macOS 10.7 Lion) regardless of the keys used for booting.
If you are unable to access the High Sierra installer through Internet Recovery Mode, then you will need to do two upgrades.....the first one being to upgrade to macOS 10.11 El Capitan which will require you to download the El Capitan DMG installer archive file from Apple.....probably easiest to use another system with actual access to the Internet & transferring the DMG file to the Snow Leopard iMac. Once you install macOS El Capitan, then you should theoretically be able to install macOS High Sierra (unless Apple changed the App Store access and no longer permits El Capitan access).
Here is an Apple article with instructions for downloading & installing older versions of macOS such as macOS 10.11 El Capitan.....make sure to read all of the instructions for El Capitan very carefully as there are multiple steps involved to extract the real El Capitan installer into the Applications folder:
How to download and install macOS - Apple Support