It's in the North Sea.
Azkaban has existed since the fifteenth century and was not originally
a prison at all. The island in the North Sea upon which the first
fortress was built never appeared on any map, Muggle or wizarding, and
is believed to have been created, or enlarged, by magical means.
Pottermore - Azkaban
Cornelius Fudge identifies it as being in the middle of the North Sea
‘… I’ve just come from Azkaban,’ Fudge had panted, tipping a large amount of water out of the rim of his bowler hat into his pocket. ‘Middle of the North Sea, you know, nasty flight … the Dementors are in uproar –’ he shuddered ‘– they’ve never had a breakout before.
Harry Potter and the Half Blood Prince: CHAPTER ONE — The Other Minister
Although that would seem to conflict with the fact that Sirius swam back to shore from it. We're also advised that it's to the south of Hogwarts.
I swam as a dog back to the mainland … I journeyed north and slipped
into the Hogwarts grounds as a dog … I’ve been living in the Forest
ever since … except when I come to watch the Quidditch, of course …
you fly as well as your father did, Harry …’
Harry Potter and the Prisoner of Azkaban: CHAPTER NINETEEN – The Servant of Lord Voldemort
Given that Hogwarts is known to be in Northern Scotland, the implication is that Azkaban is somewhere off the coast of Yorkshire, Durham, Northumberland or possible southern Scotland.