I recently had to reset my computer to factory settings. I didn't bother making a back-up of my Oblivion intallation, since I assumed I could just re-install it.

Only problem is ... I have the GOTY Edition, and my copy of Disc 2 seems to have disappeared off the face of the eath!

If I install only Disc 1, would I be able to play Vanilla Oblivion, without SI or KotN? Or would the GOTY Edition not work without Disc 2 at all?

Disc 1 should just be the vanilla game. So yes, it should be no problem for you.

Disc 2 would be where any additional info/content would be stored.