In The Hobbit, Bilbo Baggins has a sword called "Sting" that glows blue when it detects the presence of orcs/goblins.
When Gollum killed the last goblin nearby, it stopped glowing blue.
How fast does information about the presence of goblins spread to the sword, and how is the information transmitted?
Sting appears to be constantly aware of the presence of Orcs; consider in The Hobbit, for example, after Bilbo has fallen down into Gollum's lair:
It shone pale and dim before his eyes. "So it is an elvish blade, too," [Bilbo] thought; "and goblins are not very near, and yet not far enough."
The Hobbit Chapter 5: "Riddles in the Dark"
And then later, when he's following Gollum out (and presumably getting closer to where the goblins live):
Painfully he got up, and sheathed his sword, which was now glowing faintly again, then very cautiously he followed.
The Hobbit Chapter 5: "Riddles in the Dark"
Or in Fellowship, when the Orcs pass them by in Lórien:
A little way off he heard a harsh laugh and the tread of many feet on the ground below. There was a ring of metal. The sounds died slowly away, and seemed to go southward, on into the wood.
[...]
[Frodo] drew out Sting: it flashed and glittered like a blue flame and then slowly faded again and grew dull.
Fellowship of the Ring Book II Chapter 6: "Lothlórien"
It appears as though Sting is just an antenna that responds to the presence of Orcs. The exact frequency of updates is impossible to determine, owing partially to a lack of data and partially to a lack of understanding of how Sting functions. Sadly, you're going to have to accept that the answer is "as fast as it needs to."1
This also makes it difficult to answer how that information is transmitted. The simple answer is "magic," but we're not given any information about how that magic works. Presumably it detects something unique about the Orcs, but what that might be is unknown.
1 Alternately, "as fast as anybody thinks to check"
