Large mercenary companies like the famed Italian ones, or the state-backed ones that many German principalities sponsored were less common in the Industrial era post-Napoleon for sure, for the reason mentioned–large conscript armies and line infantry systems meant that such companies typically could not operate at scale to be decisive in a great power war.
However many European/Western powers continued to make use of people we’d identify as mercenaries. They were often called “Scouts” or similar terms. Some were employed by governments, some by companies. I know you said you didn’t count the native armies working for the East India Company, but the British South Africa Company definitely employed paid “Scouts” that were functionally mercenaries. There were similar arrangements with several American companies in the American west, and in the American industrial region many large industrial concerns employed mercenaries as strike breakers.
Rulers in Asia, both India and Southeast Asia / China, made use of mercenaries intermittently during this time as well.
In terms of open warfare, mercenaries of this time tended to kind of blur some of the traditional definitions. A traditional mercenary company was organized by a private leader, or sometimes a state that would organize a company and hire out its services. Mercenaries in the 19th century tended to be individual fortune seekers who would travel to conflict areas and offer to enlist. Often times they weren’t paid any special amount over other enlistees, but they were really there more often than not for chances to plunder and etc.
Several American “adventurers” of the era did this–Henry McIver for example fought for 18 different States/entities in his life. Often times for whatever reason these soldiers of fortune would be given much higher rank than they probably ought to have, so that may have been part of their motivation. Take American Thaddeus P. Mott. He left home at 17 and received a commission as a lieutenant fighting with Garibaldi’s men in Italy, later he fought for the Union Army in the Civil War as a member of the New York Volunteers, attaining the rank of Lieutenant Colonel (he was involved in suppressing the New York City Draft Riots.) He then went to the Ottoman Empire, that signed him as the equivalent of a major general (a rank probably far above his knowledge/competencies.) Later in life he became a military recruiter for the Egyptians. In each case he was enlisted/commissioned in the service of foreign princes, he wasn’t technically a mercenary, just a member of their military. But he was mostly pursuing these opportunities to make his fortune (usually through means other than the salary paid.)
You also have the Prussian Prince Felix Salm-Salm, educated in the Prussian military, he offered his services to the Union Army in the Civil War. Valuing the prospect of a foreign, professionally trained officer, he was brought in as a colonel of a volunteer regiment. He was breveted a Brigadier General by the end of the war and fought in several battles. After the war he went to Mexico to offer his services to the Emperor Maximilian, who eventually made him his aide-de-camp. He returned to Prussia to participate in the Franco-Prussian War, in which he was killed. Mercenary? I don’t know. Again, he wasn’t on a mercenary contract, he was commissioned in the proper military of the United States and later Mexico, but he obviously didn’t have any strong “personal allegiance” to those countries.
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. He was one Napoleon’s “Croats” (in his case actually a Serb from Croatia). Mercenaries were definitely in decline in the early 19th century, but still around as with