An article in British History Online , as illustrated by this John Zephaniah Bell painting says:
“Here [Westminster Abbey/Sanctuary/Cheyneygates] the unhappy queen [Elizabeth Woodville] was induced by the Duke of Buckingham and the Archbishop of York to surrender her little son, Edward V., to his uncle Richard, who carried him to the Tower, where the two children shared a common fate.”
Ashdown-Hill’s The Mythology of the “Princes in the Tower” (ch.9, p.49) talks about the confusion between Shrewsbury and Sir Richard Grey, who WAS arrested at Stony Stratford. ch.10 p.54 includes your c19 portrait: “Buckingham was also a leading member of a delegation which, on Monday 16 June, was sent by boat a short distance up the Thames to Westminster Abbey, to try to persuade Elizabeth Woodville to release her younger son, RICHARD DUKE OF YORK, from sanctuary and send him to join his elder brother at the King’s Lodgings in the Tower. However, the person who actually led the deputation into the sanctuary at Westminster had to be a priest. Therefore the group was led by another royal cousin, Cardinal Bourchier, Archbishop of CANTERBURY”.
The same volume points out that we don’t know about any “common fate”, whilst putting us in a better position to find out.