How can I effectively time a short thawing and brining of a a turkey?

I can't remember how many times I've remembered to take out the turkey much to late for a full proper defrost. Yes, you can probably thaw it in cold water. But I would go a step further if you're brining it anyway. Alton Brown posted a blog post yesterday with exactly the same problem as you. Apparently Professional food people forget to defrost their birds too.

The solution (see what I did there) is to thaw the bird by unwrapping it and submerging in a brine contained in a large bucket or cooler or other food safe vessel, covered and tucked away in a closet or garage or … wherever. I slap a probe thermometer in the brine with an alarm set to go off if the temperature of the solution rises above 40 degrees F. That said, I typically go with a 2 day soak and have never had an instance where that temperature has been reached. By the time the bird is thawed, the brine has done it’s job (two jobs actually) and I’m ready to roast.

In short, if it's still frozen, I'd probably just toss it straight into the brine and let it sit for 2 days. (Assuming you have some place that can keep it under 40F/4C until cooking day). The brine will defrost the bird much quicker and brine the meat simultaneously. One less handling of the bird. It's what I've done every time I'm in this predicament of trying to defrost a bird in time to cook.

Edit: To emphasize Jefromi's comment. He's absolutely correct. Especially if you place the turkey in a brine in a cooler and put it in your garage/basement, etc... you definitely need to make sure your brine stays between 0C (32F) and 4C (40F). The easiest way to do that is to put the turkey in the brine and periodically monitor the water temp, adding ice to drop it back to the safe range. A thermometer that can warn you would be ideal. If you manage to find enough space in a fridge that would certainly be safest, but I assume most people don't keep enough space for a turkey in a pot.