Standing in line at the checkout counter at my local supermarket a few days, I noticed one of the trashy and salacious rags with a cover story about the Duchess of Cambridge being infertile. Balderdash. So the question arises: can an adopted child succeed to the throne.
The answer is an unequivocal NO!
The Act of Settlement (1701), promulgated by Parliament, states that the succession to the throne is limited to the Protestant descendants of the Electress Sophia of Hannover. An adopted child would not be a descendant of the Electress Sophia. There are instances where British aristocratic families have adopted issue, but these children do not have succession rights to the titles, and do not bear courtesy titles.
If the Duke and Duchess of Cambridge do not have children, Prince Harry will be heir presumptive to the throne.