The ring was only as powerful as its owner

[The Saucepan Man]

Probably the One Ring awoke when Gandalf threw it in the fire in Frodo’s house.

Ah, but it was "awake" when it slipped from Gollum's finger, hoping to be found by Goblins.

Personally, I like Underhillo's theory. It works for me that the powers of the wise to sense the use of the Ring derived from the knowledge and intentions of the wearer, rather than the Ring itself. As for Gandalf not detecting its use in Mordor, he may well have been unable to do so once it passed into its maker's domain.