It is said budgetary restrictions made for example Lützower and Brunswickers chose black as their uniform colour. Since no state would provide funds for the equipment they brought old uniforms or civilian clothing and re-dyed them black as 'the only option available'. Quality overall a very secondary concern.
Problem: which dye and processes did they use in the first place?
Go today into an eco-friendly clothing store, demand "black" and get the "you stupid crazy sinner" look. Most often accompanied with assertions like 'Black is a problem for the environment, a modern invention like aniline-black only developed by chemists during the 19th century, 'natural colour dye for black not possible'.
From a theoretical point of view
- with for example oak, acacias, walnut and chestnut your colour will be a very dark brown, but not a true black.
- other colours present before the black dye will still shine through a bit (modern how-tos for modern dyes recommending to counter a brownish tone with admixing navy blue to your modern black)
- yarn dyeing is inherently much superior to garment dyeing ('dyed in the wool')
- iron-gall black ink-like dyes will stain your clothes black, but also put a heavy toll on the fabric: tannic acid and the formation of sulphuric acid will dissolve woolen fabrics and makes cotton brittle (ink corrosion) (this inference may be quite faulty, as e.g. dyeing black with Terminalia bellirica is used in India not for colour but for durability PDF)
- carbon black (soot) pigment based dye washes out when wearing and cleaning clothes
On the actual techniques used before synthetic, pigment and reactive dyes of the modern era I only found ethnographic research of traditional processes in tribal cultures (like Maori, India).
For cellulosic and protein fibres I infer that some tannic-acid process would be used, but perhaps combined with carbon pigments.
Or rather exotic ingredients probably not suitable for saving a penny ~1800:
Natural black dyes
One important black natural dye is Logwood (Haematoxylum campechianum) which is also known as Campeachy wood because it was discovered by the Spaniards on the bay of Campeche in Mexico. It is still used today for dyeing silk in deep shades on an iron tannate mordant. It also gives excellent depth and fastness on most natural and synthetic fibres (Knecht et al., 1933). Tannins are further important sources of black dyes. Pomegranate rind contains the hydrolysable tannic flavogallol, which combines with iron salts to give deep blacks.
— M. Clark: "Handbook of textile and industrial dyeing. Volume 1: Principles, processes and types of dyes", Woodhead: Oxford Cambridge, 2011.
In medieval times only iron-gall produced a real textile dye, at exorbitant cost, while soot based colouring was also available but impermanent; really more like a fraud than a dye. (— Michel Pastoureau: "Black. History of a colour", Princeton University Press: Princeton, Oxford, 2008. Which notes a few advances over time but is almost silent on actual processes and the prevailing style of technique during the inquired timeframe.)
So, while black is an all-time classic garment colour, information on the actual dye and process used in Europe around 1800 seems very hard to locate. From what was available as circumstantial information it would also seem that true dark black colour would be a very complicated achievement. As an additional info bit it would therefore be interesting to actually see 'how black' and durable (concerning the fibres and colour) those garments then became.
A roughly contemporary and evidence based as well as theoretical treatise summing up the options and issues comes from the intriguing personality of — Edward Bancroft: "Experimental researches concerning the philosophy of permanent colours: and the best means of producing them, by dying, callico printing, &c. : vol. 1, T. Cadell, jun. and W. Davies: London, 1794. (p291ff: "Of the Black Dye, and of common writing ink as connected therewith" archive.org)
In there we find a list of issues. Among those already mentioned, that sulphuric acid prevents moulding of dyed cloth but also "rottens" it (emphasising that non-sulphur dyes increase vulnerability to mould), we also see that imported Campeche wood as well as also imported aleppo-galls from Quercus infectans bring best results — but a price issue, deeper blacks require pre-dyeing with some dark blue, over-all complicated and long running multi-stage processes, etc…
An actual contemporary description or modern analysis of the process used and results achieved (while watching price and practicability) would be a superb answer.
How did Europeans dye their garments black up to around 1800, especially regarding uniforms?