I have the same issue with the exact setup , HP DV3-4333ej with radeon 6300m.
HDMI port working on windows 7 manufacturer old driver.
HDMI port NOT working on windows 7 amd new driver.
HDMI port NOT working on windows 10.
The problem is occurring because AMD drivers for Win10 are developed for hardware that supports "dynamic mode" and our most old ladies video cards do not have it.
The solution lies in an own driver AMD and exchange of Radeon video card manager. The default software is the Radeon Catalyst and it is he who is giving the conflict in older laptops to migrate to Win10, it has to be replaced by Crimson beta.
This driver does not come in automatic updating, have to get there at DRIVERS + DOWNLOAD CENTER and to search through the "manually select your driver" and the version that is doing the HDMI output back up and running properly is the Radeon software Crimson Edition 16.1 Beta.
I hope it helps you too, here again as it was in Win7.@