Your shared shopping list looks unchanged until you open the app. It then catches up immediately. That pattern is a reason to investigate background activity, not proof that the account has stopped working.
Google’s Pixel guidance says Data Saver limits most background mobile-data use while allowing Wi-Fi and active apps. It also provides an Unrestricted data exception for individual apps. Names and menu locations can differ on other Android phones.
Compare one ordinary task
Note whether Data Saver is enabled, which connection you are using and what you expect to happen. For a shared list, a useful test is a harmless sample item added from another device you control. Do not use a time-critical alert as your experiment.
Compare the same task over Wi-Fi and mobile data, allowing the app its normal update interval. Check the information inside the app separately from the notification. An update may have arrived without a visible alert, or opening the app may trigger a fresh download.
Make the smallest useful change
If the observations point to Data Saver, try its unrestricted-data setting for that app alone. Repeat the comparison without also changing battery and notification settings. Record whether anything improved; remove an unnecessary exception rather than leaving every app unrestricted.
If the delay remains, inspect the app’s sync options, notification permission and battery restrictions as separate possibilities. The first setting you notice is not automatically the cause. Keep track of what you changed and restore settings that did not help. On a managed phone, check with the administrator if the relevant controls are unavailable.