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.

This bug is reported by many people in the Google issue tracker. It looks like a wide range of Huawei devices is severely impacted by the latest Play Services update to version 12.6.73. Apparently something changed on the Google side and they are currently investigating what is happening.

The issue is tracked in the following bug:

Authorization Failure in Google Maps Android API v2 (Google Play Services 12.6.73, Huawei devices)

Star the bug and follow messages from Google engineers.

UPDATE 2018-05-15

Some messages from the bug

Have just got off the phone to Google and Huawei. Huawei seem to be admitting responsibility. They told me there is an EMUI update due in 1 to 2 weeks that should fix "all google maps issues".

Also [email protected] states

Google has been able to reproduce the problem on a physical device and is now focusing on addressing the root cause through either a fix or other mitigation.

It looks like Google and Huawei have to look into this issue together in order to provide a permanent solution.

UPDATE 2018-05-16

Google posted the following message in the bug

In collaboration with Huawei, Google engineers have identified the root cause of this issue as an unexpected change to the filesystem permissions. The issue can impact any Android app on a Huawei device which renders a Google map.

Our engineers are preparing a new version of Google Play Services which should resolve the problem. It will be targeted to devices running Android 7.0 Nougat, or higher. We will post another update with the status of the rollout.

Thanks for your patience.

UPDATE 2018-05-17

Finally good news from Google

We are pleased to report that a beta version of Google Play Services is ready. We believe this version mitigates the issue on Huawei devices running Android 7.0 Nougat, or higher. An important caveat is that each time the device is rebooted, you will need to wait about one minute before launching the affected applications.

Starting today at midnight (Pacific Time), the beta will be rolled-out to a segment of the community that has signed-up for the Google Play Services Public Beta Program. It will ramp to increasing segments of the beta audience over the next few days. We invite you to join the beta program at https://developers.google.com/android/guides/beta-program to obtain an early release of this and future versions of Google Play Services. Once beta testing is complete it will be rolled out in stages to all users.

UPDATE 2018-05-18

As a follow-up to our last update (comment#342), we believe this issue is fixed in Google Play Services version 12.6.85. If you are still encountering the problem, please ensure your device has this version installed.

As previously noted, there can be a delay (usually less than one minute) between the time the OS is finished booting and Google Play Services starts. If you launch one of the affected apps during this window, it may not render a map. If this happens, re-launch the app after a minute or so. Subsequent launches of the app should succeed every time.

FINAL UPDATE 2018-05-22

As of 10:00 AM (PDT) on May 21, 2018, Google Play Services version 12.6.85 was pushed to 100%, following the initial beta rollout. We believe this resolves the Google Play Services issue on all impacted Huawei devices.

Huawei is working on a longer term solution which is beneficial to its users and developers.

We apologize for this incident and the impact it may have caused. We are conducting an internal investigation of this issue and will make appropriate improvements to our systems to help prevent future recurrence. Stay tuned for an Incident Report.

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