This looks to be the right family. Mary Ann born Houghton le Spring in 1871 census, but born Newcastle in 1881 census. But there is a baptism in Houghton le Spring in 1862 that fits this family.
1871 census, New York, Washington,
John Reily, 40, Labourer, b. Ireland,
Mary Reily, wife, 27, b. Ireland,
James Reily, son, 13, b. Brancepeth,
Mary A Reily, dau, 9, b. Houghton le Spring,
Elisabeth Reily, dau, 7, b. Houghton le Spring,
Thomas Reily, son, 5, b. Houghton le Spring,
John Reily son, 2, b. Gateshead,
RG10/4994/4/2
1881 census, “Back of the Ship” Wrekenton,,
John Riley, 50, Labourer to Masons, b. Ireland, Galway,
Mary Riley, wife, 49, b. Ireland, Roscommon,
James Riley, son, 22, Coal Miner, b. Brancepeth,
Mary A Riley, dau, 18, b. Newcastle on Tyne,
Thomas Riley, son, 15, b. Newcastle on Tyne,
John Riley son, 12, b. Wrekenton,
Mary Riley, dau, 7 months, b. Wrekenton
RG11/5041/42/31
1891 census, West Boldon,
Patrick Charlton, 30, Colliery Labourer, b. N. Seaton, Northumberland,
Elizabeth Charlton, wife, 25, b. Houghton le Spring,
plus children,
John Riley, Father in law, S, 62, Coal Miner, b. Ireland.
RG12/4162/12/17
A couple of baptisms at Houghton le Spring St Michael (Catholic so Latinised names)
Maria Anna Riley, born April 1862, baptized 25 May 1862. Parents Johannis Riley and Maria Meecham Riley.
Elizabetha Riley, born 1 March 1864, baptized 9 April 1864. Parents Johannis Riley and Maria Machan Riley.
Means that Marys maiden name was Meecham / Machan.
Alan.