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You will need to goto a 3rd party website that hosts Google apps and download the newest (for your device) version. The one I use is http://gappsandroid.blogspot.com/ , you can download EVERY single Official Google app.
You could use OpenGApps. I believe it is sanctioned by Google and allows you to easily select a version for your device architecture and Android version, customizing the package by which apps/services you want.
Once downloaded, you can flash the package via recovery. It is also possible to extract the apps for a manual upgrade; see my answer here for details.
After changing the Play Service version, and installing on my phone, android studio gave me a warning saying that I can't update the existing application, and I must uninstall it.
its give you warning because in you device you have signed APK and you are trying to update it using debug apk. so generate signed apk(with same certificate whatever you have used previously) then update app in device.
So, can I still release the new APK in the developer console, and will it cause issues?
in above process there is no issue then you can upload same signed apk on Google Play Store
assume you know what is signed & debug apk and how to generate both
EDITED:
Gradle is a build system. i am not able to explain it in simple way but you can read more detail of Gradle in this answer and for more about build system and Build System Overview
I suspect that what happened here is that your signing key doesn't match. (I.E. if you downloaded your release-key version from the Play store while using the debug key with Android Studio) Changing Play Services should not have this impact.
Recently I reset my ONE M7 and I found that Play Services can't be Updated .Do anyone know how to fix this.. on Play Store it won't show any update button.. I never Saw these Things in my Other 2013 phones like Galaxy S4.