when the US/UK/China, etc. sneezes, Japan/Germany, etc. catches cold

when (something) sneezes, (something else) catches a cold

When a person, group, or entity has a problem or experiences a negative situation, a related person, group, or entity will consequently have a worse problem or will experience a more negative situation. When Paris sneezes, Europe catches a cold. Terrorism in France affects security throughout the continent.
See also: catch, cold
Farlex Dictionary of Idioms.

when the US/UK/China, etc. sneezes, Japan/Germany, etc. catches cold

or

when the US/UK/China, etc. sneezes, Japan/Germany, etc. catches a cold

mainly BRITISH
If you say that when a particular country sneezes, another catches cold, or catches a cold, you mean that what happens to the first country has a great effect or influence on the second. And when the American economy sneezes, the City of London catches cold. As they say, when America sneezes the rest of the world catches a cold.
See also: catch, cold, japan, UK
Collins COBUILD Idioms Dictionary, 3rd ed.
See also:
  • if (something) catches a cold, (something else) gets pneumonia
  • if (something) sneezes, (something else) catches a cold
  • japan
  • pneumonia
  • sneeze
  • UK
  • when (something) catches a cold, (something else) gets pneumonia
  • when (something) sneezes, (something else) catches a cold
  • when A sneezes, B catches a cold
  • an insult to one is an insult to all