Here is a Walter, not even though this one does not state which county he was born in in Ireland . the second appears to be the same person, but his mother was remarried then so put both in, grant you I don't know if he is whom you are looking for but infor fits
Walter J Donnelly
Age: 5
Estimated Birth Year: abt 1886
Relation: Son
Father's Name: Michael
Mother's Name: Elizabeth
Gender: Male
Where born: Ireland
Civil Parish: Bedford
Ecclesiastical parish: St Thomas
Town: Leigh Lancaster
County/Island: Lancashire
Country: England
Registration district: Leigh
Sub registration district: Culcheth
ED, institution, or vessel: 4
Household Members: Name Age
Michael Donnelly 39
Elizabeth Donnelly 25
Walter J Donnelly 5
Joseph Donnelly 3
Mary F Donnelly 7/12
Walter Donnelly
Age: 15
Estimated Birth Year: abt 1886
Relation: Stepson
Mother's Name: Elizabeth
Gender: Male
Where born: Houghae, Ireland
Civil Parish: Leigh
Ecclesiastical parish: St Thomas
Town: Leigh
County/Island: Lancashire
Country: England
Registration district: Leigh
Sub registration district: Culcheth
Household Members: Name Age
Henry Walter Lindsay 48
Elizabeth Lindsay 34
Walter Donnelly 15
Joseph Donnelly 13
Mary Francis Donnelly 10
Thomas Donnelly 8
Ellen Donnelly 7
Hopes this helps

Joyce