I think you're over-dramatising this?
If it was the case that the driver (who was that??) did stop and exchange details and a claim has been made and settled (and you have a paper trail or can get one to show this) then you stand a very good chance of defending the charge. You need to position it that you weren't driving, state who was (it can't be two people!) and a narration of events. However, the non receipt of two sets of paperwork is a bit hard to substantiate - but not so much if there is evidence of this being a regular occurance: This is usually done by showing some sort of (written) exchange with the Post Office.....have you done this?
There's too many holes in the depiction of events to comment fully - others will add things, but it is not sufficient to plead NG and just turn up wihtout anything to upport what your state happened: It's hard to see how the other driver was so distraught at his/her error for invisible damage that resulted in a claim you allowed your insurer to pay out on? You need to review all the circumstances/facts - BUT the FtF stands either way and is a stand-alone offence.