Vance had blamed Henry Nowak's death on the "mass invasion of migrants" - but Lammy says it "has got nothing to do with mass migration".
The scheme aims to tackle the "growing challenge" of young people not in employment, education, or training.
Craig and Lindsay Foreman's son urges the UK to explore a prisoner exchange to secure their release.
The Defence Investment Plan is now due to be released ahead of a Nato summit early next month.
Harriet Sperling and Peter Phillips wed with King Charles, Queen Camilla and other royals attending.