EMP from a space nuclear blast would take out quite a few civilian satellites. Most if not all US military satellites have been hardened against EMP for a long time. The DSCS-III satellites that I used to operate had nuclear shielding in place. They even tested this by exposing a test article to an underground nuclear detonation. The first DSCS-III was launched in 1982. This isn't exactly new technology.

A space-based nuclear detonation probably wouldn't take out all of the civilian satellites either. First of all, most of them have been shielded to an extent to handle nasty solar effects. Furthermore, I doubt the EMP would be able to take out the satellites on the opposite side of the Earth from the detonation. If I'm right, the Earth itself would act as a shield for some of the satellites.

One bad thing that a space-based nuclear detonation would do is increase the background radiation level in LEO for some time, especially in the van Allen belts. Those satellites that routinely pass through the belts would be exposed to more radiation than they were designed to handle, possibly causing premature failure.

This isn't just a matter of speculation. We have actual experience with this. Back on July 8, 1962, the US detonated a large nuclear warhead 400 kilometers over Johnson Island in the Pacific. The test was called Starfish Prime. The warhead was 1.4 megatons, far bigger than any Pakistani warhead. The EMP caused power outages across much of the Pacific. The increased background radiation was said to be the cause for the premature death of the UK's first satellite. However, other satellites weren't all wiped out and we did go on to launch a bunch more over the years. Buzz may be telling a good yarn but it doesn't match the results from an actual test. He overstates the effects for dramatic effect, IMO.

If anything, the EMP from a space-based nuclear detonation would have more serious consequences for ground systems than most satellites. The systems most effected would include electrical grids, computers, and unshielded communications infrastructure within several hundred kilometers of the detonation. In other words, if Pakistan fired a nuclear missile straight up and detonated the warhead, they'd be doing themselves and their immediate neighbors the most damage. That's not a very smart strategy, IMO.

Nuclear Weapon EMP Effects

More on EMP effects

Here's some congressional testimony on the topic.