I would like to uninstall the default camera app and instead use the FOSS Open Camera. And I would also like to access the camera from the lockscreen.
How can I do this in Android 6?
Default camera is a system app, which requires root to uninstall default camera app. You can use Titanium Backup like app to remove system app (N.B: Removing system app will be harmful to your device).
Settings->Security, under device security, tap on settings icon near screen lock. Select Lock screen shortcuts. Tap on any left or right default icons, prompts to choose action: tap on select application. Tap on Apps on top, and select your required camera app(These setting is on Lineage 14.1 android nougat)
While uninstalling the pre-installed camera app can be difficult as others have mentioned, it should be relatively easy to change your default camera app to Open Camera (or whatever camera app you prefer). I prefer Open Camera as well, so I also wanted the same thing.
To make Open Camera the default (or any camera app other than the pre-installed version), follow these steps:
Caveats: sometimes when either camera app updates, you have to do the process again. Usually it doesn't happen too often, only when big changes occur to the apps. Hope that helps!