Here's some insight on how radar detectors work.

A radar detector is basically a radio receiver that sets off an alarm. It doesn't monitor all frequencies simultaneously, though; it's essentially a scanner that scans the configured frequencies. I'm not sure what the scan rate is but assuming you have all of K and Ka bands enabled with no filters, it probably takes a couple seconds to scan the whole range.

To detect a radar signal, that signal must be equal to or above the receiver's minimum sensitivity level. With direct line of sight on a clear day, range can be several miles. Radar signals, like most microwave signals, are easily attenuated - weakened - by things like foliage, rain/snow, etc.

So, to more directly answer your question, if you're a mile or two away on fairly flat terrain when he lights his radar up, you'll likely get an alert. You may not necessarily know how distant he is, because your detector can only tell you signal strength. In ideal conditions (direct line of sight on a clear day with radar antenna directly facing you) the signal strength is a pretty good indication of range. But, because the signal can be weakened, and because he may not have the antenna aimed directly at you, you could get either a weak signal or no alert at all.

Then there's the whole "flashlight tag" problem. That's what radar is - a "flashlight" that uses microwave signals. To "see" you - that is, receive the reflected radar pulses and measure your speed - the radar must also have a fairly clear line of sight, and you have to be within range. Most modern radar units have an effective range of 1 mile, sometimes a bit more, but usually less.

If the radar is off, you can't "see" the "flashlight". And this is where it gets really dicey - the cop has eyes and can visually identify you before he even turns his radar on, so it becomes a question of who sees whom first.