The Quiet at the End of the World by Lauren James.
 
How far would you go to save those you love? Lowrie and Shen are the youngest people on the planet after a virus caused global infertility. Closeted in a pocket of London and doted upon by a small, ageing community, the pair spend their days mudlarking and looking for treasure - until a secret is uncovered that threatens not only their family but humanity's entire existence.
 
Now Lowrie and Shen face an impossible choice: in the quiet at the end of the world, they must decide what to sacrifice to save the whole human race...