OP, read the process pl. This applies to you and would avoid you posting questions which raise issues which will not arise.
London borough of Islington are claiming that if the address is wrong with dvla then this is our own fault, not theirs (understandably) but if anyone has any insight or advice, would be much appreciated!
Yes, stop discussing matters with the council*! They are obliged to follow the set procedure which is that if they wish to pursue you** for the penalty then they must register the penalty with the Traffic Enforcement Centre, send a statutory notice to you to this effect - known as an Order for Recovery- and tick one box on the enclosed form(Statutory Declaration(SD) - don't worry about the name of the form, civil servants who devise these procedures love such things)- sign and get countersigned(together) and return to TEC.
So, at present as regards the council your role is passive i.e. wait for the OfR.
*- EXCEPT, as one notice has gone missing it's always possible that a similar fate could befall the OfR. I suggest you notify the council of your correct address for correspondence to ensure that on this point you and they are singing from the same hymn sheet.
But as regards this forum, we need to make the most of the time between now and the OfR (at least 2 weeks, probably more like 4+) to get facts from you to enable you to submit reps against the PCN when the new one arrives.
And yes, you are correct that anyone could claim to have not received a PCN by post. So what? The reset process would send another and another and another...so what's to be gained by telling porkies?
And of course 'It is a criminal offence to make a false statutory declaration.'