Who was next in line?

Henry VII married Elizabeth of York, daughter of Edward IV.

Since her two younger brothers were kept in the Tower of London by her uncle, Richard III, and never were seen again, this left Elizabeth as the premier heiress, and Henry VII made a prudent marriage with her, thus uniting the Lancastrian and Yorkist branches of the family.

Their dynasty was a short one, though, lasting only three generations and a little over 100 years

Upon Richard's death, Henry VII immediately sent his men to Sheriff Hutton (where Richard had looked after many royal children, his own, including illegitimate ones, and members of his brothers' families), who took them all back to London. Young Edward, George's son, was put into the Tower and kept there as a prisoner for about ten years.It was said he was simple minded; whether he was perhaps Down's we don't know...maybe he just went mad in captivity. Henry eventually executed him age 19 or 20 on trumped up charges of either attempting an escape or 'talking with Perkin Warbeck.'

Henry also had 'a bastard son of Richard III' put to death for contacting someone in Ireland (where there was considerable support for York.) This was probably John of Gloucester.

Regarding a claim, to the throne, George's son would have been under his father's attainder but this could have been repealed.

Many nobles had a better claim to the throne than Henry...he is thought to have only been about 17th in line (some say even further down the line than that!)

He was kept a prisoner by Henry VII who had unified the House of Lancaster and the House of York. He was executed in 1499 after an allege escape attempt.