Greywolf1: The most common causes for this kind of problem I'm aware of are mods and incomplete deinstallations. In your case it doesn't look like a mod problem. To rule out an incomplete de-installation, I recommend to uninstall Oblivion, and then to delete whatever may be left of the installation manually. This is the game folder, the Oblivion folder in My Games (described in the second post of this thread), and the registry files. The registry leftovers are the main culprits.
To get rid of undesired registry leftovers is not really difficult, but bears the risk to damage the registry and make the PC ununsable, and should therefore be done only by people who know what they are doing.

You mention that "GOG logs me as having played for 1 Minute". Am I right to interpret this as "Galaxy logs me as having played for 1 Minute"? If this is true, Galaxy might be another potential problem cause - but I'm not using Galaxy, so I couldn't help you with that.

Perhaps you try the following approach:
- uninstall Oblivion (and all mods) completely,
- make sure that there is nothing left (no game folders, no registry entries),
- download and install the "direct download" version of Oblivion from GOG,
- and try again.
If Oblivion works, you may try again with the Galaxy version, or continue to play the non-Galaxy installation with or without additional mods installed, which may help you to identify the real problem cause if necessary.
Otherwise perhaps someone else can help you (sorry).

Good luck!