After the first three Carthusian priors were executed on May 4, 1535, the next three leaders in line, Humphrey Middlemore, William Exmew and Sebastian Newdigate, were executed on June 19 that same year and these two monks were taken from London to the Charterhouse of St. Michael in Hull. In the wake of the Pilgrimage of Grace, Rochester and Walworth were tried in York by Thomas Howard, the Duke of Norfolk and found guilty of treason.
The Catholic Encyclopedia offers this detail about Blessed John Rochester:
Priest and martyr, born probably at Terling, Essex, England, about 1498; died at York, 11 May, 1537. He was the third son of John Rochester, of Terling, and Grisold, daughter of Walter Writtle, of Bobbingworth. He joined the Carthusians, was a choir monk of the Charterhouse in London, and strenuously opposed the new doctrine of the royal supremacy. He was arrested and sent a prisoner to the Carthusian convent at Hull. From there he was removed to York, where he was hung in chains. With him there suffered one James Walworth (?Wannert; Walwerke), Carthusian priest and martyr, concerning whom little or nothing is known. He may have been the "Jacobus Walwerke" who signed the Oath of Succession of 1534.
So if he had taken the Oath of Succession in 1534, Blessed James Walworth must have recanted his oath or refused the Oath of Supremacy in 1535.
It must have been an agonizing death--hanging until death by exposure and dehydration--left like their brother Carthusians in Newgate Prison in London as this site notes:
The image above does not depict the Carthusians hanging from the battlements; it is from the Wikipedia entry for Blessed John Rochester. It is by Vicente Carducho and is part of a series of paintings in the Charterhouse of El Paular near Madrid. The article on John Rochester includes these details about their trial and execution:
The two London monks were brought from Hull to York and brought before the Lord President of the North, the Duke of Norfolk, on trumped up treason charges. Condemned to death, they provided the desired menacing spectacle for the city when on 11 May 1537 both were hanged in chains from the city battlements until their bodies fell to pieces.
Blessed martyrs of the Carthusians, pray for us!