Visually, 1 second for every 186,000 miles, because that's the speed of light.
A geostationary equatorial orbit (GEO) is a circular geosynchronous orbit in the plane of the Earth's equator with a radius of approximately 42,164 km (26,199 mi) (measured from the center of the Earth). A satellite in such an orbit is at an altitude of approximately 35,786 km (22,236 mi) above mean sea level.
Because it's only 22,236 high, even if you severed it at the tippy top, they would notice pretty much instantly (or near to) if they have a clear view. (I assume they'd be able to look through a telescope). (source)
But, Because of the slow rate of rotation of the Earth (only one revolution per 24 hours) the cable has to be very long - theoretically at least 25,000 miles, and in practice closer to 60,000 miles. (source)
Still, even at 60,000 miles, it will take less than a second for the light to travel and show us that it has happened.
Now, non-visually, the effects are going to depend on HOW you handle the severing. Is it a large explosion? Is a line mechanism just cut?
Let's also talk about how elevators are handled in large buildings. Most of the time, the elevators go to certain floor, then you have to change elevators to go up the rest of the way. For safety's sake, I would expect it would be handled this way. So maybe 1/2 way or a 1/4 of the way up, the doors open and you go to another elevator in the same bay, switching sides as you go.
If the elevator doesn't work, you'd be stuck at the second to last elevator up. Now, some of these proposed space elevators don't work this way--they are a single elevator that travels up at a high speed, but this is the way I'd build it, mainly because segmented systems can be easier to maintain. There are lots and lots of models for this, so I can't answer this question fully without knowing the method of design (there's the ski-lift version that involves multiple cars, and I'd expect that this would be segmented in some way as well).
I also think that the speed is important to the design as well. See this answer as to the top speed of the elevator. That's because if you have people travelling in the elevator for 2-3 days, those people are going to have NEEDS. Like eating and going to the bathroom. Will there be a bathroom on the elevator? Will the elevator(s) have stops on tiny platform to allow for eating and drinking? How large will each one be? Is it one single car, or several sent in a row? If it's several, the car that doesn't make it might procedurally send a message down on arrival, and that will be noticed, or the system simply won't take them up any further and they might radio for help.
Answering this question can only be fully accomplished once you've done a lot more research on how these things work and how your operating system is going to work. This determines how communication works, how many stops there are and all that--since I am sure the ground likely communicates with the elevator(s) along the way. If many are being sent up, one right after another, that changes things vs. one single car, (because that will take 2-3 days to reach the top) as will stops and communication with people on the top.
