Oats will not give you nitrogen credit, as clovers would.  Oats would help to soak up, and hold any carryover nitrogen that is in the soil.  So, if you have applied more nitrogen than is utilized by the wheat crop, oats would help to pick up at least some of that nitrogen.  BTW, the radish will not produce nitrogen either.  The radish will scavage any nitrogen that is in the soil, but it will not create more.  Only the clover in that mix will do that.

I think oats could germinated at depths up to 3" deep.  If you spread oats and then disk-rip, I would expect that you will cover the oats with a variable depth - some oats would be covered "shallow", and some would be covered "deeper". That would be my expectation.  Whether you would cover some oats too deep to germinate,  I don't know.

I will say that your radish and clovers should be seeded closer to 1/2-1" deep, at that maximum.  That is why I have more concerns with the radish and clovers, than the oats, if spreading seed and then running a disk-ripper.

Again, if you want to try this, try it on a small scale the first year.