The sentences with 'go' don't sound natural to me. 'Go on' means continue, but 'go on with it' is not a phrase which means much. I'd understand what somebody meant if they said this, but this isn't an expression which native speakers would use.
'Get on with it' is a very natural phrase, however. It means : stop talking about it, stop worrying, stop messing around, stop wasting wasting time, and let's just do what we've got to do.