>Is there any evidence that those countries got access to US nuclear secrets?

There is no evidence that they didn't. And why qualify your question with United States? The leakage of secrets we need to be concerned about is not just from the US. If they got secrets from non-US countries, that is a concern too.

Turning back to encryption backdoor keys, other countries, besides the US, will also have these keys, if such a thing is instituted. The leakage of these keys will have consequences for everyone, in all countries, even if the keys are different for different countries. International communication and commerce is a commonplace daily activity for most of us now, so a breach of security anywhere can affect people everywhere.