The process for calculating the number of coils in a pickup is actually relatively easy. There are two steps: Determine the length of wire used, and calculate the number of coils in that many meters of wire.
The first step is to measure the resistance of your pickup. Get a multimeter and set it to measure ohms, and connect each one of its two leads to one of the two wires coming from the pickup. Be careful not to hold onto both leads, in order to keep the wires in place; remember, your body is a resistor, as is the pickup; if you hold onto the leads, you're creating a parallel resistor circuit, with your body in parallel with the pickup; this will mess up your resistance reading.
Once you've done this, cut off a good length of coiling wire from your spool of spare wire (NOT your pickup!!), at least a couple of meters' worth, and measure both the exact length and the resistance of this wire. Use these numbers to calculate the resistance per meter of the coiling wire, and divide the resistance of the pickup by this number to get the number of meters of wire in the pickup.
Now that you have the length of wire used, calculate the average size of one loop of wire. Do this by finding the circumference of a finished coil (maybe wrap the extra wire that you just cut around the pickup and measure the length of the wire segment needed to go around it), and find the perimeter of the bar magnet, which, presumably, is the same as the size of the first coil. A good approximation for the average size of a coil can be found by averaging these two numbers; I'm sure it would be possible to derive a slightly more precise formula than that, and were I feeling just a little less stressed at the moment I would be glad to derive it just for fun, but I lack the time so a straight average will have to do.
Take this average size of a coil (you did measure it in meters, didn't you? OK, probably not; if not, convert it now), and divide the total length of wire that you have by it. This will give you the approximate number of coils in your pickup.