Children of Men/WMG

Children of Men takes place in the Half Life universe.
Between 1 and 2. The Combine is suppressing the ability to reproduce already. It all fits, except there's no mention of the Seven Hour War.

Britain is not the only country with a functioning government.

 * It was possible that the British government uses the "Britain stands alone" line to say that it was the only surviving nation left.
 * Supported implicitly in the movie. The youngest person on Earth was an Argentinian who was killed (ostensibly by accident) while leaving a club in Buenos Aires. Doesn't sound like an anarchy to me. The idea seems to be that there are functioning governments, but they are a minority among those that collapsed.

The sterility plague was not caused by nuclear fallout.
If it were, humans would not be the only species affected, yet we see many animals (cats, dogs, chickens) that have much shorter lifespans than 18 years. Whatever caused it specifically affected only humans.