When using a 4.5" spinner on an OS 1.20, the spinner's adapter doesn't have enough threads to go on if retaining the original 2 nuts for the prop.

What is the best solution? to use only one prop nut and risk a loose prop if the engine backfires? or will the spinner adapter fill the role of the second/front nut?

The adapter nut takes the place of the prop nut and lock nut that comes with the engine.
If you notice, the adapter comes with a new prop washer, and the adapter nut has a shoulder that fits inside the washer when you are tightening down the prop. To understand, just hold the prop washer and adapter nut in your hand. Fit the shoulder on the adapter nut into the hole of the prop washer and you'll see what I mean............
Take the engine's original prop washer, nut and lock nut and put them back in the engine box.........use the new prop washer and adapter nut to tighten down the prop to the back plate and the engine. So far, I haven't had a problem with my 125 back firing the prop off the engine with my new Dave Brown Spinner set up. I like mine....................

But in this case there is nothing to stop the prop and spinner flying off in case of a back-fire, right? makes me worry a bit.

I saw the same engine with two nuts, the second one accepting the spinner screw, and when it back-fired the screw simply twisted and sheared off, the spinner came off, but the prope was still secured to the engine.

Don't know what else to tell ya...........the spinner adapter nut is what it is.........put on your helmet and flak jacket before you start the engine I guess........I've been wacked with props off of 4-strokes before and I survived.............now that I think about it, I believe thats when I learned to swear............................................. ....

I've got your same engine and spinner set up and like I told you.........I like mine.