After updating to EOS10 can't download photos says all four phones on account are locked

From the information I have been given, it seems your iPhones have not been trusted by our computer. If this is the case, plug in your iPhone to your computer and unlock it. A message to trust or not trust the computer should pop up, so press trust. Otherwise, please tell me what version of OS X/macOS you are running, and if EOS10 mean iOS10.

So, you are opening Photos, connecting your iPhone and unlocking it, then clicking "import" and either "import all photos" or "import selected"? And just to make sure, you are using the Photos app on OS X Yosemite or later, not iPhoto?

iOS 10 requires OS X 10.9.5 or higher and iTunes 12.5. It is not going to work on OS X 10.6.8. It's time to either upgrade, or think about using iCloud Photo Library or an online photo backup service.

I Am sorry to break it to you, but chances are your Mac will not upgrade past OS X 10.7.5. Please tell me what processor you have by going to the Apple logo at the top left, and about this Mac. That will let me know if you can update your compuer. Unfortunately, it is probable that you have to buy a new computer in order to do anything with your phone Syncing.