Yeah, but you're just compressing the nerve, not severing it. You're not completely cutting off the signal, you're just interrupting it enough that your brain can't figure out what's going on downstream of the pressure. Why does it take a while? I'm not actually sure why. It may just take that long for the nerve, and/or the tissues around the nerve, to compress enough.
Nerves do weird things sometimes, and the brain is even weirder. Like you can make your whole lower arm instantly go dead for a while if you whack your funny bone (aka the ulnar nerve) just right, and it doesn't always have to be that hard; but you can put sustained pressure in that exact same spot for a long time before you start noticing any difference at all, let alone full-blown pain or numbness. If you ever hurt your back bad enough to pinch a nerve (which I've done twice), very often it takes several hours after the injury before you start experiencing problems. In both my cases it took all the way until the next day before the nerve problems kicked in, even though the disc was pressing on the nerve root the whole time.
While I'm not sure why it works that way, I'm glad it does. Getting a massage would be a very disturbing experience if it didn't.
But like I said, bloodflow can be a factor. If you cut off the blood supply to a nerve it will most definitely start freaking out, stop functioning properly, and start to die just like anything else in your body. A lot of people think that it's cutting off a major blood supply to the limb that causes the numbness, though, and that's not really what's going on. If that were the case the limb wouldn't just go numb, it'd start turning funny colors too. And that's not even mentioning the other problems that would go with it, like Compartment Syndrome, blood clots, and the possibility of having a stoke. It also wouldn't explain other factors that can cause the same type of numbness: sustained vibration, certain auto-immune diseases that attack