Its awe inspiring to see tall chimneys here in the UK, and to think it was hand built by men in the 1800's. How did they make these structures, over 200 feet tall, by hand and keep them perfectly straight?
Barrow Bridge Chimney comes to mind, but there are many, many more in the UK.
In short, they were constructed in the only way possible, the same way any massive construction was; from the ground-up, on a strong and stable foundation, by skilled bricklayers, generally standing on a lot of scaffolding built alongside by skilled carpenters, laying one brick at a time.
I'm afraid 'advanced bricklaying techniques' is quite far from my area of interest, so I hope someone else will fill you in on that.
If you're interested in the theory and engineering behind it, as well as common issues such as how to straighten a chimney that's started leaning (!) and how to prevent damage from lightning strikes, check the literature links below.
As for the scaffolding, which is what made it possible to reach those heights, the most common solution was to "simply" build scaffolding all the way up to the top during construction, and then tear it back down. It was almost always made entirely out of of wood and nails, especially in the early Victorian era, and at least as impressive a construction as the chimney itself. It's easy to forget scaffolding needs to carry not only the weight of the people and equipment at the top, but the bottom level of scaffolding needs to support the weight of all the levels above as well.
I've seen a few photos of "bottle oven" kilns being constructed without external scaffolding and I'm not sure if it was internal or just a ladder (see here for instance). However, for tall smoke stacks, scaffolding was always kept as basic as possible, and depending on what was easier to build and tear down, it could be internal or external. Here are some examples of chimneys, a couple of water towers and a lighthouse:
First stages of the Mount Morgan mine chimney in 1905
Halfway up the Graves lighthouse 1904
Construction of the Hook Norton brewery smoke stack, 1880's
Malmö (Sweden) watertower, 1915-16
Kalmar (Sweden) watertower 1898 and during deconstruction of scaffolding.
Relevant literature:
Tall Chimney Construction – A Practical Treatise on the Construction of Tall Chimney Shafts by Bancroft & Bancroft, 1885.
Chimney Design and Theory by Christie, 1902.
