I have the Kindle app on my iPhone and iPad. When I go into a book on the iPad and add a highlight I never see it appear on the same book on my iPhone, even if I press the sync item from the side menu on the iPhone.
I was able to fix this by removing the book from my iPhone and redownloading it.
Now if I want to see highlights sync immediately between the devices I have to go to the Library after adding a highlight and then go to Library on the other device, press sync and go back into the book. Then a few seconds later the highlight appears. Not ideal but it works.
I had highlights on an iPhone which were refusing to sync. Manually triggering a sync from within the Settings area and killing/restarting the app failed to fix it.
I restarted the iPhone (hold power and home until you see the silver, Apple logo appear) and then opened the Kindle app again.
I could see a lot of network activity (the icon was cycling next to the 4G symbol at the top) and then all of a sudden the highlights all synched and were visible on https://read.amazon.co.uk/notebook (also https://read.amazon.com/notebook ).