One way is to serve the update yourself via your own server. If you have this setup on your existing app in the play store, awesome. Otherwise you have to have to push an update to the old app ASAP (through the play store) that is compatible with your current game server, that will contain this check on your apk server.
You will have to check the server version, and if it's been updated, fetch the updated app. Also add some code that checks if the apk server is providing a binary or redirecting you to another url. If it's the latter, you can have your app open a browser to open that link (as to why you want to do this, read on.)
Let it settle down a bit (2 days should be enough), then update both your game server and play store app at the same time.
Now you can modify the apk server so that it redirects everyone to the play store link of your app.
Users that were updated through the apk server may be prompted with a new version in the play store again, though the update doesn't really add anything (AFAIK there's no way to query if an update is available in the play store through the play store itself). Also I don't know how Play Store checks for an update -- if it's just the version code and version name from the manifest, they may not be served with the update anymore (which is good).
Users that were not updated through the apk server but were able to get the patch for the server checking mechanism will now run the app and find that they'll be redirected to the playstore, of which they will now download the update.
The only limitation with this solution is that users that were not able to get the server check mechanism patch may have to figure this out for themselves that they need to update via play store. You can help them by sending a notification email regarding the update.