My simple answer is.. it depends on the property & room availability when you show up. Even as Titanium myself, if I show up and they simply don't have a room ready, I may have to wait. Or they have a room ready but I want an upgrade, I might wait for the upgraded room to become ready (because at 1 PM, housekeeping may not have had a chance to clean up after a guest who checked out at noon).

I was in a Marriott in China and they followed the rule -- only my room had privileges. Both in check in/out time as well as lounge access time.
A Marriott in Shanghai did NOT extend lounge access to my 2nd room, but DID give late check out.
A random Courtyard in NJ extended full privileges to both my rooms, giving $10 credit to EACH guest per room (so I got $40 breakfast credit each day for 4 guests).

So... it's been completely up to the property in my experience and how nice you ask, and how generous they feel.