What is the most common English term for a person who attempts a coup d'état?

In Latin America, we have the Portuguese/Spanish word golpista (from the word golpe = coup d'état). In the British media, I've read coup monger and also putschist (from German word putsch = coup d'état). But are these expressions as common in English as golpista is for Latin Americans?

There is no specific English agent noun for the leader or participants in the modern sense of a coup d'état, at least in common usage.

You will usually find coup coupled with a generic term for a leader. In fact, coup leader has the most results in COCA and the BNC among all my searches. Coup organizer is roughly equivalent though much less common. In headlines or in less formal usage, coup chief or coup head might also be suitable.

Coup ringleader and coup boss are dismissive and pejorative, as would be other negative words coupled with coup: plotter, conspirator, schemer, etc. You need not formulate these as attributives, either; you're more likely to read about the orchestrator of the coup than the coup orchestrator, for example.

One could make the case for usurper or deposer, but I would not say they are commonly applied to actors in a coup d'etat. A deposer can be one who deposes in the sense of removing another from power, but this sense has been largely overshadowed by the legal sense of deposer, one who collects a deposition (sworn testimony). Usurper in contrast is well-understood, but it carries strong connotations of the seizure of a crown (or something likened to a crown). It's unremarkable to say Henry VII was the usurper who ousted Richard III, but it's unusual to say that Nasser was the usurper who overthrow Farouk. After all, there was no longer a throne to usurp— Nasser replaced the monarchy, not just the monarch.

Putschist, which you have mentioned, is established, but its usage seems to be falling. Rarer still is coupist, which can be found in a dictionary or two—

one that attempts or supports a coup d'etat ^(MW)

— but little elsewhere.

Assuming that by coup d'état you mean the sudden, usually violent overthrow of a government outside of regular political processes, revolutionary or one of its synonyms (rebel, insurrectionist, etc.) would seem most likely. From Collins Dictionaries:

revolutionary

noun ​
social studies a person who supports or takes part in a revolution

revolution

noun
social studies a sudden and great change, esp. the violent change of a system of government

The main distinguishing characteristic of a coup seems to be that the group of revolutionaries is relatively small and may come from within the government, and that the (attempted) overthrow is very swift. English doesn't seem to have a specific way of distinguishing either a small or an internal group of revolutionaries, although I would say that some synonyms are less likely to fit those parameters.

Some examples:

In this satirical allegory, farm animals representing Bolshevik revolutionaries successfully execute a coup d'état (which they call 'The Rebellion') —first entry (Animal Farm) in Wikipedia's "List of fictional revolutions and coups"

A coup d'état is thus a revolution although it does not often result in a dislocation within the country as always happens in the case of a classic revolution. . . . [I]f the coup succeeds the constitution . . . is replaced with other laws promulgated by the revolutionaries. —Carlson Anyangwe, Revolutionary Overthrow of Constitutional Orders in Africa (also uses the phrase coup-makers)

A coup launched in Berlin by a group of radical socialist revolutionaries is brutally suppressed by right-wing paramilitary units from January 10 to January 15, 1919; the group’s leaders, Karl Liebknecht and Rosa Luxemburg, are murdered. —"This Day in History", The History Channel website

The term coup-makers, mentioned in the second example above, is sometimes used specifically for those who are most directly responsible for a coup attempt. This may be used more or less interchangeably with revolutionaries by some authors, but at least one source has argued that a distinction should be observed.

Coup leaders often proclaim themselves “revolutionaries,” but coups are not revolutions. . . . Revolutionaries seek fundamental social, economic, and political change; coup-makers may seek this, but they may act to prevent change or merely to gain the rewards of political office. —"Coup d’etat." International Encyclopedia of the Social Sciences

In practice, I believe the more general term is at least as common as any of the more specialized terms; however, it is extremely difficult to prove this, as the phrase "the revolutionaries" stands alone and only collocates with coup within the wider discussion making any automated corpus search difficult. However, coup-makers would be an excellent term to clarify the specific type of overthrow without a lot of context.