It depends a lot on where the country is, who has had influence there before and the politics of the country in question.
For example, Menelik II of Ethiopia brought in Russian advisors and bought Russian mountain artillery before the Battle of Adwa 1896. He bought some Hotchkiss revolver guns as well, but did not want French advisors - probably because they were subjugating North Africa at the time.
So, this country will be wanting advisors with good reputation from a country willing to send them, and from countries that do not have significant interests in colonising them.
In the 1850s, Britain has the best reputation for industry. Prussia and France have the best reputation for the army - although Switzerland for internal security/gendarme/police also works. Britain is good for trade and legal reform. Russia might be welcome if the nation is vast and has a lot of cavalry.
Sweden has a decent reputation for internal security, navy and shipping at this time (they did provide the advisors to form the modern Persian Gendarmerie, in essence a regular force to combat tribal forces). After the civil war, US advisors might be sought after.
The Dutch might be asked to help with trade, shipping and legal frameworks, considering their role in Japan.
So, here's your candidates;
Russia - army (especially cavalry).
Prussia - army.
Britain - industry, navy, shipping, trade, army.
France - industry, army.
USA - army, industry, shipping (after 1865).
The Netherlands - trade, shipping, navy.
Switzerland - internal security.
Sweden - internal security.