I once was able to handle one of these pots at a small museum in a Staffordshire pottery and see its use demonstrated. That particular pot was made as Flatlander described, with a tube running vertically up from the bottom, narrowing towards the top. The base of the spout was close to the bottom of the pot. The pot was turned upside down and liquid was poured into the tube, filling the whole pot and the tube up to a mark scratched on the inside of the tube, presumably calculated to avoid getting liquid into the spout or above the top of the tube once the pot was righted. As you turned the pot up the right way the liquid left in the tube dribbled out. There was not much mess, perhaps because the vacuum formed as the pot was righted held back the flow of liquid to a trickle. I think wwh's idea of a tube running up the handle would have been even less messy and perhaps some were made in that way; in my youth, many teapots had handles too hot to hold (try a silver teapot as an example!), so I don't think that would have been seen as a big disadvantage - anyway, what were servants for?

To me the strangest thing about the whole idea is that as there was no way of inserting tea leaves into the Cadogan pot, or extracting them afterwards, the tea would have had to be brewed in a separate pot and then poured into the Cadogan – you might as well have poured the tea in through the spout! By the time it was served to eager drinkers it would have been pretty cold. The usual recommendation when making tea is to rinse the pot with hot water to warm it before making the tea. I guess you could do that with the Cadogan, but probably it was just intended to be a drawing room demonstration of a simple but intriguing scientific principle rather than a serious teapot!