I have, at last, solved the mystery of loud, insistent and unfamiliar bird (kee-kee) calls from nesting birds in big trees around my garden; with occasional very rapid flights in which - silhouetted against the evening sky - I could see what looked rather like a hawk, with long curved and pointed wings; and a short curved beak.
(From where I was, from underneath, as the birds flashed-by at dusk, wings flapping - they did not look anything like the picture above - the colour seemed blackish, and the beak did not look red...)
Then, this morning, in better light (and at a time of high swooping activity) I spotted one that had, to my amazement, what seemed to be a green back! On checking a UK identification website from the RSPB, I saw the only possible bird was: a ring-necked parakeet...
On using this as my search term, plus Newcastle upon Tyne, it instantly emerged that this is a known-thing; that ring-necked parakeets have been nesting and breeding in urban Tyneside for at least five years; and that they are spreading and increasing. It has been in the local newspapers.
So, that is the answer. It seems bizarre, but now I know the answer to the mystery birds is... obvious. Sort of.