As of now Event Gallery supports Google Photos. You can use private and public albums without any issue. But since you can't modify existing albums anymore you may want to use protected albums as well. In order to do this you need to open the components options. There you can hit thebutton to allow Event Gallery the access to your photos. Afterwards you can use the normal album selector to chose your albums from your Google Photos account. Keep in mind that you can only use one Google Account right now.
In case your images do not appear in the front end, you can try to delete the folder
/cache/com_eventgallery_picasa manually or by using the Joomla! cache manager. This will trigger a reload of the data. Keep in mind that every request to Google Photos is cached for some time to avoid a slow website. If you still see small or empty XML-files in that cache folder, you might have specified an invalid Google Album.
You can also check the log data too. In the back end of Event Gallery you find the menu item
allow_url_fopen in order to let your server talk with Google or the Token-Refresh-Server.
Since the communication with the Google token service and Google is using HTTPS your server needs to support this kind of communication. If something fails you should see an entry in the log file mentioned above.