According to the church's website:

https://whitefriarstreetchurch.com/saint-valentine/

In the early 1800s, restoration work was being carried out in a Roman church when the saint's remains were discovered. In 1835 an Irish Carmelite called John Spratt visited Rome and spoke to the people. The Church was so taken with Spratt that one of the gifts bestowed on him were the remains of St Valentine from Pope Gregory XVI. They were received by Archbishop Murray of Whitefriar Street Church in 1836 but eventually went into storage. In the 1950s/60s, they were discovered and brought back into the church, as St Valentine's Day was quickly becoming an annual event for lovers, and group weddings became common in the last couple of decades.

Regarding the bones of St Nicholas, they're interred at Jerpoint Abbey in County Kilkenny. They were brought to Ireland during the crusades for safe keeping.