hi
wi-fi does not "time out" for Notifications.
Notifications works with either a persistent wifi or cellular data connection.
Settings/Notifications have you set it so that Notifications Show On LockScreen?
this is the only thing i can think of. sorry if it doesn't help.
you can verify you have a wifi connection from Lock Screen by swiping up to get a look at yr Control Panel (if you have Control Panel to be accessible from yr Lock screen)
HOWEVER: THE FOLLOWING IS AN APPLE SUPPORT DOCUMENT
it may be indicating that if your particular wifi connection is using a proxy server, then it may not be able to connect to the apple server for notifications
https://support.apple.com/en-us/HT203609
quote
Unable to use Apple Push Notification service (APNs)
When a device is connected to Wi-Fi and does not have a cellular data account or a cellular data signal, push notifications are not received.
Devices using APNs need a direct connection to Apple's server. If a device is unable to connect using cellular data, it will attempt to use Wi-Fi if available. If there is a proxy server on the Wi-Fi network, the device will not be able to use APNs, because APNs requires a direct and persistent connection from device to server.
When connecting to APNs, iOS devices will use the cellular data connection if it's available. Only if the cellular connection is not available or viable will the device switch to Wi-Fi for APNs connections.
endquote