The answer ended up being quite simple; just had to add the following line to my updated notification:
.setTicker("Upload Complete")
Here's the complete code:
Builder uploadedBuilder = new NotificationCompat.Builder(context)
.setSmallIcon(R.drawable.logo)
.setContentTitle("Upload complete")
.setTicker("Upload Complete")
.setContentText(Constant.totalUploadedImages.get()+" " + uploaded_count + " uploaded")
.setAutoCancel(true);
Notification uploadNotif = uploadedBuilder.build();
mNotificationManager.notify(Constant.NOTIFICATION_UPLOADING_ID, uploadNotif);
Constant.totalUploadedImages.getAndSet(0);
}