Which country was the first one , in which the Christian Church was separated from the state? I know it must be in Europe, but I don't know which one exactly.
It is unclear from your question what do you mean by "separated from the state".
In fact European history knows only one theocratic state - the area under direct papal control centered in Rome, known at different periods as Papal states or Vatican.
The rulers of all other European states were secular persons.
Still the laws of many European states empathized Christianity as the state religion or Christian god as the source of power.
In all European states the church performed some duties that now viewed as the duty of state. For example, the registration of births and marriages. But among population of different confessions these duties were performed by different religious bodies. For example, births and marriages among Jews were registered by Jewish religious organizations.
I will list some events that may be relevant to your question.
In the years 1103 to 1075 there was a dispute between the Pope and some of European monarchs, notably, the Holy Roman Emperor over who has the right to appoint church officials such as bishops. Traditionally they were appointed by the emperors but the dispute led to the outcome that this power was limited.
This event can be seen as making local church hierarchy somewhat independent from the monarch. The political struggle between sovereign monarchs and the church lasted for many centuries with monarchs either trying to limit the Church power, or to take control over church themselves as it happened in Britain and Russia.
In 1144 the population of Rome revolted in an attempt to establish a secular government, known as Roman Commune. The revolutionaries expelled Pope and declared allegiance to the Holy Roman Emperor.
As a result of Protestant reformation and the Thirty Years War it became impossible for many states to maintain allegiance to a single confession. In many countries the laws were modified so to reflect the multiconfessional character of the population. This mostly touched the Holy Roman Empire.
The Great French Revolution of 1789–1799 was marked with boldly anti-Catholic character. Started with removing the church privileges and trying to subordinate the church to the state (as opposed to separate) by making all clergy the state employees it proceeded to an attempt to replace Christianity with some secular cults. This somehow followed the way of reforms by Peter the Great in Russia who abolished the patriarchal chair and made the Church essentially a state ministry by putting a secular ober-procurator in the head of the church.
Thomas Jefferson of the United States in a private letter for the first time declared that the church should be separated from the state in 1802 but this declaration had no legal force. Still some US courts later cited this letter in the rulings.
France in 1905 declared the state separated from the church by law.
The Russian revolution of 1917 followed the case by declaring the church separated from the state, essentially rolling back the reforms by Peter I. The patriarchal chair was restored, and the state launched massive atheist propaganda.
It seems the first state declared officially atheist was Albania, where in 1977 production, distribution and storage of religious literature was made a criminal offense.
The first country with a separation of the Christian church from the state was the Roman Empire.
First, trivially, before Constantine the Church was persecuted by the government and was entirely separate.
After that, there were incidents that showed the Church could claim power above the government. Consider Bishop Ambrose of Milan in the late 4th Century:
Theodosius I, the emperor of the East, was excommunicated by Ambrose for the massacre of 7,000 persons at Thessalonica in 390, after the murder of the Roman governor there by rioters. Ambrose told Theodosius to imitate David in his repentance as he had imitated him in guilt — Ambrose readmitted the emperor to the Eucharist only after several months of penance.
The Emperor of Rome at the time was all powerful in the secular field but was reduced to submission in this instance by the Bishop. This did not happen in every instance, most Bishops were not as firm as Ambrose but it did set a precedent for the later Popes to use.
The first government in the world to separate church and state and to offer religious toleration was the American colony of Rhode Island in 1636. It was started by Roger Williams as a "hedge or wall of Separation between the Garden of the Church and the Wilderness of the world." His ideas would inspire John Locke and later Thomas Jefferson.
The first national government to officially separate church and state was the United States with the US Constitution. In 1776, Virginia, South Carolina, Pennsylvania and Rhode Island had disestablishment enshrined in law. The last US state to separate church and state was North Carolina in 1875.
The second nation to obtain a separation of church and state and religious tolerance is likely Brazil in 1891. Mexico and France had anti-clerical revolutions before this time obtaining short-lived separations of church and state through a repression of religion. In Mexico:
In 1859 the Ley Lerdo was issued—purportedly separating church and state, but actually involving state intervention in Church matters by abolishing monastic orders, and nationalizing church property.
Catholicism became the state religion of France again in 1803 re-established by Napoleon following the French Revolution starting in 1789.
Since the OP asked about Europe, specifically: I didn't check every nation in Europe, but I assume that Switzerland is likely to have the earliest and freest laws governing religion. Complete religious toleration was granted in 1874 but separation of church and state was rejected in 1980 by ballot initiative. Here is a good paper on the development of the role of religion in Europe. Religious toleration seems to have been common in Latin America by this time.