The index on Scotland's People lists Mary as Mary Stewart Ward so maybe the couple never married. As you say the list allows limited search criteria.
In 1901 the only James and Mary Ward I can see (with James a coal miner) are living at New Cumnock, Ayrshire, as below but there is no Thomas and the age of Mary is out
Edit and in 1891 they had no Thomas
WARD, James 38 Coal Miner b Irvine, Ayrshire
WARD, Mary 37 bIrvine, Ayrshire
WARD, John G 10 b Irvine, Ayrshire
WARD, Jeanie H 8 b Irvine, Ayrshire
WARD, James 6 New Cumnock, Ayrshire
WARD, Mary H 3 b New Cumnock, Ayrshire
WARD, John Brother Married 34 Coal Miner b Irvine