According to the Downloading the Expansion Files section of this document →
It says that it is common that your .obb file will not automatically download.
When your application starts, look for the expansion files on the shared
storage location (in the Android/obb// directory).
A) If they are there just continue loading the application.
B) If they are not there then do these two things:
Perform a request using Google Play’s Application Licensing to get your app’s expansion file names, sizes, and URLs.
Use the URLs provided by Google Play to download the expansion files and save the expansion files. You must save the files to the shared storage location (Android/obb//) and use the exact file name >provided by Google Play’s response.
Note: The URL that Google Play provides for your expansion files is unique for every download and each one expires shortly after it is given to your application.