My problem started with a cat. She loves enforcing my laptop breaks and having some laptop time herself. Whenever that happens, I lock my laptop and let her be for 5 minutes. Without fail she'll manage to turn off wireless and enable the darn Windows Narrator. And no, turning off the Narrator shortcut doesn't help.
Issue here is that lock screen works under completely different environment and permanently disabling narrator shortcut for your user will do nothing. No, solving this requires a bit more interaction and the easiest way I found is through registry editing.
To turn off the Narrator, there is a well documented WinEnterLaunchEnabled registry value. For our user we can find this at HKEY_CURRENT_USER\Software\Microsoft\Narrator\NoRoam but for logon user this hides at HKEY_USERS\S-1-5-18\Software\Microsoft\Narrator\NoRoam. Setting this DWORD value to 0 sorts the issue even without going for restart.
However, since my cat also plays with touchpad, I decided to remove the whole Ease of Access portion to ensure erroneous touchpad movements as cat lays down cannot turn anything on. For this we need the BrandingNeutral value. This can be found at HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows Embedded\EmbeddedLogon. Setting DWORD value to 8 followed by a restart sorts out that issue.
Those full of trust can download registry file with these settings here and import them automatically while others can do the registry changes manually. In either case Windows 10 will become a bit more cat friendly.