First I cut off the narrow bottom end of the cylinder to remove the tube that goes inside the bottle, then I cut off the top tube that runs to the spout of the pump. That let me push the plunger out through the bottom.

I did the same thing with the other pump, except I had to get the entire tube out of that one so that the only part that was left was the fitting that screwed onto the bottle. To do that, I ended up cutting it and tearing it out with pliers. It would have been easier to use a drill to drill out the center, but then I would have had to leave the room where I was sitting.

Once that's done, all you need to do is join them together.