Hi, and thanks for joining our community.
A few points of interest.
There is general agreement that the first showband single was Tom Dunphy and the Royal Showband's "Come Down The Mountain Katie Daly" released in 1962. However, it did not reach number one.

However, the band's next single, "Kiss Me Quick," an old Elvis Presley song, performed by Brendan Bowyer, reached number one on the Irish Charts in August, 1963 and stayed there for seven weeks, becoming the first showband single to top the Irish charts.
(Information source: 1994's "The Golden Era of Irish Showbands" published by the Limerick Rock n' Roll Club and edited by Paddy Brennan.)

By coincidence, Brendan and the Boys also had the second number one hit with "No More" in December of 1963. Dickie Rock and the Miami knocked them out of the number one position with their single, "There's Always Me," a week later.
Another interesting fact: Eileen Reid was the only female showband singer to have a number one hit in the 1960's with "A Fallen Star," in May, 1964.