I've never had a coggins take 6 weeks; usually 7-10 days is more like it. However, ask your veterinarian prior to scheduling it, so you know how long it should take. I've gotten most back within the 7 day window, and the longest I've had one take is 14 days. I have gotten some back as early as 3 days. I think it will depend on where you live, and how far you are away from the nearest test facility.
It is required for most shows, yes, but also if you ever plan on moving your horse out of state, or even to different boarding facilities, etc.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."