Where does the story take place? Ireland or England ?

Frank O'Connor's story "My Oedipus Complex" is most likely set in Ireland, although an outright statement of location is not mentioned at any point. Much of the story focuses on themes of a small-town poor family in post-WWI Europe as experienced by the five-year-old narrator, so exact setting is not extremely important for an overall comprehension of the story.
However, there are a number of clues surrounding names, streets, and landmarks that imply that the story is set not just in Ireland but specifically in Cork, the town where O'Connor himself was from. Rathcooney Road, where the narrator mentions wanting to play, is in fact in Cork. The church mentioned near the beginning of the story is also in Cork.

After breakfast we went into town; heard Mass at St. Augustine's. . . .

The religious Order of St. Augustine came to Cork around 1300, finishing a large church on Washington Street in 1781.
Furthermore, many of O'Connor's stories are autobiographical. While he was actually an only child, the author's relationship with his not-readily-employed, often abusive, and alcoholic father was strained. In the story, the narrator's father's name is "Mick," a shortening of the name Michael, which was O'Connor's actual father's name and a particularly Irish style of nickname.
http://www.staugustinescork.ie/

It doesn't explicitly say so anywhere in the story. Nevertheless, it's pretty safe to assume that it's set in Ireland. The author, Frank O'Connor, was Irish, and the story contains a fair number of autobiographical elements. A mention is made of someone called MacCarthy, a recognizably Irish surname. Reference is also made to attending Mass at St. Augustine's church and visiting the friend of Larry's mother at the convent, Mother St. Dominic. Ireland was, and is, an overwhelmingly Catholic country, so this provides a further hint as to the story's setting.
But the clincher comes in the following line, where the boy Larry is trying to get his father's attention:

"And after that I want to go up the Rathcooney Road," I said loudly, afraid I’d forget something in all those interruptions.

Rathcooney Road is in the city of Cork, the place where none other than the author of "My Oedipus Complex," Frank O'Connor, was born and raised.