Let me start by saying I am not an expert and I do not work for BT. That being said:
At face value, on the strength of the information provided:
My first thought is that when BT sent a new phone, they presumably sent one that was identical to the one you already possess. Although unlikely, if there was some kind of manufacturing defect that runs through that particular model of phone - or even throughout one particular batch of that model of phone - it is conceivable, just, that both phones could be suffering from the same defect. By way of experimentation, and inasmuch as you seem to have exhausted most of the possibilities, I would suggest trying a different phone altogether - not necessarily a new one, but just any phone with Caller Display that's been proven to work elsewhere ( maybe borrow a neighbour's or friend's phone just for the sake of the experiment? ). Alternatively - or additionally - if you have, or happen to know anyone who has, an old stand-alone BT Caller Display unit ( such as the CD1000, for example, but anything similar would do ), try plugging that into the same socket, both with and without a phone also connected, to see if it makes any difference. Although admittedly cumbersome, these experiments could possibly help to narrow the possible cause of the problem.
Did you make any other changes at around the time the problem manifested itself? - In particular ( although I'm guessing not, since the phone is cordless, but still ) did you add any new phones to the circuit elsewhere in the property? If you pick up the phone and dial 17070, what message do you get?
If all of the above yields no new results, and it is categorically determined that the problem lies outside your property ( notwithstanding BT's assertion that this is the case, which for the reasons above may not necessarily be conclusive ), then regrettably you have no other option than to pursue BT themselves to resolve the matter. I do not say this lightly - one thing of which I do have considerable experience is 'sagas with BT' - but if the fault is not anywhere in your home, it must be in their equipment, somewhere. If - and I do say 'if' - there is or ever was a 'national problem with Caller ID', it seems surprising that it hasn't come up before this; and if in fact it is a national problem, you'd think BT would indeed have addressed it long ago. Frankly, if you first became aware of the problem last summer and it was October before they told you there was a 'national problem', I'd be sceptical; BT are not in the habit of spending their money unnecessarily, and how many cases like yours would there have to have been, involving extensive and expensive involvement of Openreach, before they declared it a 'national problem'...?
At any rate, just my humble opinion ... but if none of the above yields anything new, it seems to me you have three options:
1 ) Pursue it with BT until they solve the problem, one way or another;
2 ) Try switching providers, although that in itself can admittedly be a minefield and there's no absolute guarantee that it will solve your problem. You could make your situation known to any potential new provider, but ( see my comments elsewhere on the subject of guarantees ) it's highly unlikely that they would give you any iron-clad guarantee that the problem would not exist with them - and I personally wouldn't trust it if they did.
3 ) Learn to live with it. Again, I do not say this lightly - we've had to learn to live with a sub-standard phone service, faced with a similar lack of options and after a similar saga ( only partly involving BT in this case ), whereby the incoming volume on our phones is literally the lowest I've ever experienced in my life. However, beyond a certain point, the official line will inevitably become 'we've done all we can; if you think you'll get a better service elsewhere, by all means give that a shot' - in which case, see option 2.
Sorry this isn't more cheering, and others may well have more optimistic views on the matter; but if you do decide to carry out any of the experiments above, and / or you have any further information, by all means post it and we'll see if it gets any closer to solving the problem.
