I don't think it's reliable. I got my approval letter yesterday (dated 7 Nov) and the website still says awaiting decision. I can't book my appointment yet as the HO letter hasn't reached the council yet
Call your Local council again and ask them to get in touch with the HO, also make sure you give them your HO Ref number, as i have called my local one today even it's too early for me to get any reply from HO (Basiccaly was just trying my luck) and the lady from the council had to ring me but i couldn't pick so she left me a voicemail stating that:
To ring or email them back with these details
IF I have received Citizenship invitation from HO:
-Full name
-DOB
-HO Ref number
So this way if they need to call the HO and ask them whats happening, and at least they have all the above details that they need about you when contacting HO!
Make sure you insist when calling or give them a quick visit! Good luck