Dr. Bashir clearly has an English accent (and his father has a distinct Cockney accent), but do we hear in canon and/or EU material where on Earth he comes from? (Picard has an English accent but is French, so there's only so much we can draw from accents).
Bashir's personnel file in the non-canon Star Trek: Starship Creator game states that he was born in London, England.
He was also implied to have ties to London in the DS9 novel, "Hollow Men".
The Centaurian Embassy took up the whole of a row of what had once been four tall, terraced town houses. Sisko and Garak materialized in a small flower garden at the front of one of the houses, at the foot of a set of wide stone steps that led up into the buildings.
It was late evening in early autumn. The air was damp and smelled of leaves. Twilight. Looking out from where they stood, with the embassy behind, there was a quiet street running parallel to the row of terraces. Over the road, running left to right, was a line of trees and then a row of lamps that went along the length of the embankment. Beyond the thread of lights, Sisko could see the dark line of the river.
"Where are we now?" Garak said, staring up at the gray wash of the sky, and then taking in the outline of grayer buildings that spread out along the far bank of the river.
"London," Sisko said.
"I think I may have heard the doctor mention it."
"Well, this is certainly more his part of the world."
Star Trek: Deep Space Nine novel, "Hollow Men", by Una McCormack
Bashir's place of birth was never established within canon, however, according to his Memory Alpha page, DS9 writer and producer, Ronald D. Moore, said that his family's roots were "probably in England", hence his father, Richard, speaking in a London accent when he appeared in the episode, "Doctor Bashir, I Presume".
"In my mind, Julian was of Sudanese (like Sid), Indian, or Pakistani extraction, but that the family's roots were probably in England, hence the accents." (AOL chat, 1997)
In the real world, there are a large number of families of South Asian descent living in England.
Some of Bashir's hobbies are also arguably clues as to his British background. For example, he and O'Brien were regularly shown playing darts at Quark's (DS9: "Prophet Motive").
Bashir also liked to roleplay as a British secret agent -- obviously inspired by James Bond -- in Quark's holosuites (DS9: "Our Man Bashir").
GARAK: I take it your character is some kind of rich dilettante with a fascination for women and weapons.
BASHIR: Actually, my character is far more disreputable. I'm a spy.
GARAK: A spy? And you live here?
BASHIR: Yes. I work for one of the nation states of this era, Great Britain, which is battling various other nations in what is called the Cold War. This apartment, my clothes, weapons, even my valet were provided to me by my government.
GARAK: I think I joined the wrong intelligence service.
Star Trek: Deep Space Nine - S04E10 - "Our Man Bashir"
According to the Paramount-licenced game Star Trek: Starship Creator, Julian Bashir's place of birth is London, England, Earth.