Here on Scilly, the generally, or entirely, absent list of otherwise common mainland birds is a long one:
Others have mentioned Magpie; also Green woodpecker - I hesitate to include Great-spotted, because we've had several this year, including one that bounced along in front of me at the Higher Moors. They're more of a scarce visitor I think.
We also have no Jays, Coal Tits, Yellowhammers, only the occasional Reed Bunting; Dabchick caused a mini-twitch earlier this year; no Great-crested Grebe; no regular owl species, though a Long-eared Owl or two typically overwinter; Jackdaw is an occasional visitor only (I've had just the one in 9 months); likewise Hooded Crow; Red Kite is also an occasional visitor - there have been a couple of birds this year, both of which I've missed, and the numbers sometimes found in Cornwall do make their absence something of an abiding mystery.
There are also some weird population dynamics: Mediterranean Gulls are more common here than Black-headed Gulls; Sandwich Terns are common in summer, but Common and other terns remain scarce, or entirely absent; Eurasian Shag comfortably outnumber Cormorants, though perhaps that one does at least make sense with the preponderance of rocky islets; and Great-Northern Divers remain common, whilst other divers remain very scarce.