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You can recover photos from a non-functional mobile device in one of three ways depending on how the phone was set up.
iCloud Photo Library.
If you had iCloud Photo Library enabled on the device, all of the photos on it should be mirrored to your iCloud storage space in the cloud. Simply signing into your iCloud account on your replacement device using the same ID as you used on your old device and enabling iCloud Photo Library (Settings > Photos & Camera) should cause your device to start downloading all your photos.
If you don't have a replacement device, you can always access your photos through a web browser at iCloud.com
My Photo Stream
If you had My Photo Stream enabled on the device, photos taken in the last 30 days should be mirrored to your Photo Stream space in the cloud. Simply signing into your iCloud account on your replacement device using the same ID as you used on your old device and enabling My Photo Stream (Settings > Photos & Camera) should cause your device to start downloading all your photos. Please note though that the 30 days is calculated from the current date, not the last time you took a photo.
Back Up.
If you don't have iCloud Photo Library enabled on your device, your photos should be included in your back up (providing you have your back up enabled)
If you opted to back up through iTunes the photos in your camera roll are automatically backed up for you, if you opted to back up to iCloud you have the option of turning the back up of photos in the camera roll off (Settings > iCloud > Storage > Manage Storage > Backups > This Device > Backup Options)
Assuming you do have a back up, your photos can be returned to your mobile device by undertaking a restore.