Same here. Launch window comes up, I click "Play", launch window disappears, hourglass cursor is visible for a second, disappears - nothing. Oblivion task is not shown in the task manager.
Report to GOG support.

To all Oblivion players: Can you run Oblivion today?

My OS is Win10 64 bit, by the way, and there has been a Windows update yesterday indeed. Has someone managed to play Oblivion on a Win 10 PC today?

Update: What's going on here? Oblivion runs on Steam, but not on GOG. Nehrim runs (how can I check which Oblivion installation Nehrim uses? - all I see is that it uses the normal oblivion.ini in My Games\Oblivion).
Tried to run the GOG installation again. Settings have gone (they were there when I started writing this post), re-entering them in Launcher - Options ok, click "Play", launcher window disappears, reappears, click "Play" again, Oblivion starts as if nothing had happened.
Has launching either Nehrim or Oblivion (Steam) affected the change?
There is still something wrong: The launch window appears twice before the game starts. This is reproduceable.

To GOG Support (assuming you read this): I reported the problem to you not long ago. Please update the problem ticket with the update to this post. Thanks.

@Eldrac: Please check your Oblivion installation. Do you still have the problem you reported?

Yeah i got it working by making it window mode in oblivion.ini file but this is not a a real fix only a temporary solution.

I had the opportunity tonight to test Oblivion on a different Win10 64bit PC - and everything worked fine.
Whatever the problem was / is, it doesn't appear to be a general problem with the OS (or something happened meanwhile to fix it) - nor with the game itself.
I will try again tomorrow on the first PC, let's see what happens.

Update: No change on the first PC: Launch menu comes up twice, then Oblivion starts normally. Unless someone comes up with an idea why this happens, I see no way to get this resolved. Different behaviour on 2 different PC's, but nothing preventing me from playing - not a reasonable basis for investing a lot of time and effort.