Photostream only syncs photos over Wi-Fi, make sure you are connected to Wi-Fi to begin with.
Photostream only syncs when the camera app is closed, ensure it is closed.
Photostream only syncs when the battery is above 20%, try recharging the device
Try disabling photo stream on your device (settings > iCloud), restarting your device and then re-enabling photo stream.
If this doesn't help you may need to reset photo stream. You can do this at icloud.com by clicking on your name in the top right corner and then the advanced settings in the pop-up dialogue box that appears.
Note: disabling photostream and re-enabling it will result in any photos that are currently in the photostream album on your device, that are older than 30 days old, not being added back.
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