This seems similar to my setup.
viewtopic.php?f=21&t=14947The route I went was to gut the console, and rewire the switches to the new hardware. This is probably the easiest and fastest, but you would lose the old organ sound. In either case, you would need ribbon cable/wires and connectors appropriate for your boards. It looks like the switches would be wired directly to the spreader boards that you have purchased. My experience is that it is unlikely that you will damage the switches by working on them. If you find a switch isn't working, it probably wasn't working from before. Gutting it also has the advantage of opening up space to mount circuit boards to, and make connections easier.
For safety's sake, a tip is to turn on the organ when it isn't connected to the wall. This will discharge any capacitors (which are typically present in the amplifier) and make the organ safe to work on.
If you are planning on gutting it, a question would be if you would like to use the existing speakers and amplifier. While this was possible in theory for my setup, I chose not to do it because I felt I wouldn't be happy with the results (For sampled pipe organs to sound good, I feel you need a lot more power and clarity than those old sounds systems were able to put out). I also didn't want to fiddle with getting the amplifier working with a PC input, which would have been very time consuming.
As for the wiring of your hardware, there should be wiring diagrams available from the supplier you bought the hardware from. They will also tell you how to wire the key and stop switches to the board. Sometimes they are a bit of a puzzle to figure out, but if there is no other circuit connected to the switch and you wire them incorrectly, it just won't work properly (you won't damage the hardware - at least that was my experience). You may need a multimeter to figure out how the switches work.
Some have gone the route of adding midi to the existing organ, and not gutting it (an option if it works well already, or you have access to the original repair manual and don't mind puzzling out how to get it working). The only trick is that some organs feed the sound signal through the keys (instead of just using them as switches as your midi board would do). If this is the case for yours, you may have to either use a relay system (which from my research appeared to be very expensive), or find a way to make it that the midi board can sense the switch activation in spite of the extra current going through. I've read of others applying a current to the keys to neutralize them. This all seemed too complicated and time consuming for me, so I didn't bother with it. If your system has a nice hammond organ sound though, it may be worth keeping (even though I prefer real pipe organs, I also like the hammond sound - especially if you have a rotatory speaker).