REST Serialization escapes URLs
Actually, I can't see how this could have been different prior to 8.3 - PHP's json_encode seems to have been escaping / by default forever, and neither Drupal nor Symfony seem to have overridden that in the past.

The service ID and class you would need to override is serializer.encoder.json (Drupal\serialization\Encoder\JsonEncoder), adding a JSON_UNESCAPED_SLASHES to the options bitmask in the constructor. (Confusingly, there's a separate service named serialization.json, which isn't the one that's used here.)

However, note that \/ is a valid escape sequence in JSON (if not strictly required), and a JSON parser must interpret this as a literal forward slash character to be working correctly. There should be no backslashed URLs in the parsed values. Is it possible that your client's parser library has a bug?