I have a few of my favourite games stored with Gog, but after a recent re-build of my Anno 1404 after a catastrophic corruption after downloading Anno 1404 history version, I'm trying to back track and reload the version I had on Gog Galaxy,. Gog galaxy now will not open and therefore I have no access to any of my games. I've tried re-downloading Galaxy, and running it as administrator, but nothing seems to work
Any assistance in getting it up and running again would be greatly appreciated.

OzBoz

You can launch installed games without galaxy by starting the executable in the game's installation folder directly. You'll miss out on GOG Galaxy bells and whistles like achievements, cloud saves, and sometimes multiplayer.

You can download owned games using a web browser. Hover over your username at the top, pick Games, click a game from the library page, then ignore the big blue download button and instead expand the Offline Installers section below that. Try moving C:\ProgramData\GOG.com\Galaxy\logs to the desktop, then make an attempt to start GOG Galaxy. Check the newly created folder for fresh logs and see if they offer a clue as to what's wrong.
Other than that, open a support ticket, if you haven't already. Tell the chatbot the issue is of type "other" twice to skip to that step.

Thank you very much for the response. I have eventually gained access to Anno 1404 no thanks to Galaxy, which still isn't working. I am now anticipating the same sort of problems with my other games stored on Galaxy. Is there any way I could transfer all my games to Steam or even my PC? I really would now like to abandon GOG altogether, it's just too much trouble.

There are no games "stored in Galaxy". They're installed on your computer. Like I said,
1. You can play installed games directly by going to the install folder and launching the game executable.
2. You can use a web browser to download offline installers for your games. Unlike GOG Galaxy, these create Start menu and desktop shortcuts that launch the games directly.

Have a look at the attached screenshots for examples. For more details, see How do I download my purchased items? - GOG Support Center

Also note support tends to be slow. If you opened a support ticket but haven't received a reply yet, it doesn't mean they forgot about it.