I'm playing Oblivion GOG version, with mod 'Skip Tutorial v3' ; it has worked with previous versions, but in this case I got to top of staircase in the initial prison, but 'E' does not open the door; it just flashes the cursor and goes back to normal. Any idea what is going on, or how to fix it?

I'm also using Unofficial Oblivion Patch V3.5.1

Wait isn't spacebar used for opening doors and E for jumping?
Also i didn't check the unnoficial oblivion patch out, i just started playing this game and went straight to the shivering isles content since that is new to me.

This is correct, both Oblivion and Morrowind use this non-standard keybind configuration as their default. You can of course change it yourself, but people likely aren't going to do that until they become aware of this.