Was the progress of astronomy in the 1800s surprisingly slow, and if so why? The 1800's was a century of fantastic scientific discoveries in chemistry, geology, biology, engineering and so on.  Is it correct to say that astronomy did not keep up with this development? And if so, why? If not, what might be some main reasons for me (and others) having this impression?

The late 1700's saw a couple of revolutions in astronomy such as the discovery of Uranus as the first non-antique planet and the measurement of solar system distances thanks to the transits of Venus. As I understand it, the next big leaps in astronomy came in the early 1900's with the HR-diagram and the "discovery" of galaxies and more. The view of the universe did not seem to change much between 1800 and 1900. Nowadays every decade has its revolution such as inflation, dark matter, dark energy, exoplanets.

Spectroscopy, optics, photography, Doppler effect, and electricity all developed greatly during the 1800's but don't seem to have brought any revolution to astronomy until later. Is relativity and quantum physics behind this new flourishing? Had the Newtonian physics reached its limit for potentially revolutionizing astronomy already about year 1800? Or was astronomy just out of fashion, maybe because of more profitable employment for the brightest brains in the age of industrialization?
I feel this question is so broad, it might well be the topic of an interesting book on the history of astronomy, should someone be inclined to write it. :)

Anyway, I think a few points could be made briefly.

1. Collecting data

In astronomy, that means observing the cosmos. That means using an instrument of some kind, typically a telescope, and gathering information through it. Telescope performance is dictated by many factors, but the most important one is size (or aperture).

Telescope size grew rapidly through 1600s and 1700s, from Galileo's 1.5 cm refractor in early 1600s, surpassing the 1 meter aperture in early 1800s - Herschel's 40 foot reflector. There was a steady stream of improvements regularly throughout that 200 year period. One could say that the first golden age of the telescope aperture race both culminated and ended with Herschel and his giant telescopes.

Then there was a lull, briefly interrupted by Lord Rosse's 1.83 meter telescope, the Leviathan of Parsonstown, in mid-1800s. Then nothing again.

The aperture race was resumed only in early 1900s, with the 2.5 meter reflector on Mt. Wilson, the Hooker telescope. Afterwards, throughout the 20th century, and now in early 21st, the race is going strong, with the 10.4 meter Gran Canarias segmented reflector currently in the lead, and the 39 meter E-ELT reflector being under construction at Cerro Armazones.

https://en.wikipedia.org/wiki/List_of_largest_optical_telescopes_historically

2. Interpreting data

The year 1900 marks the boundary between classic physics and the new physics. After that year, relativity and quantum mechanics took off. This is what enabled the new cosmology to emerge in the 20th century.

In other words, with the 1800s science, even with tons of data, there would have been no way to figure out, basically, everything. Supernovas? The expansion of the universe? Dark matter and the rotation of galaxies? This is all based on 20th century physics. 19th century physics would have been clueless.

Astronomy used classic physics to derive interpretations from data pretty quickly, and that process had achieved great success already well into the 1700s. That's when the structure of the solar system was figured out, way back to Kepler in the 1600s. Herschel found Uranus in late 1700s.

There are some exceptions here. Stellar parallax was detected in early 1800s, which enabled an estimate to nearest stars. Spectroscopy showed that distant stars are made of the same elements like the Earth in the 1850s. Around that same time, Neptune was discovered.

So the 1800s was not quite a completely dry period, in terms of theoretical progress.

In any case, a limit was reached anyway in late 1800s, because what was needed was new paradigms in physics to give new life to the interpretation process. That boost occurred after 1900, with relativity and quantum mechanics.

Cosmology is highly dependent on physics (and vice-versa).