Is there any way to see the analytics for my app (i.e. downloads)?

Yes. Apple and Google will have the most up-to-date stats for your app. You can see that information in your Developer Accounts:

• Google: https://play.google.com/apps/publish

• Apple: http://itunesconnect.apple.com

You can also use Google Anayltics to see where people are going in your app and when they're using it Real-Time: https://www.bvmobileapps.com/pdf/how-to-get-google-analytics-tracking-id.pdf