It depends on a number of factors, but I'll try to give you an idea of how long it takes for legal action to be taken in these cases. First of all, you have to file a complaint with the court. This means that you need to file paperwork with them stating your claim and asking them for compensation. This process can take up to a month or two, depending on how busy the courts are at the time. Then, once you've filed your complaint, it will be sent over to someone who specializes in copyright infringement cases (usually an attorney).
The attorney will review your complaint and decide whether or not they believe your case has merit. If they do believe it has merit (and if they think they can win), they'll work with you on preparing your case for trial so that they can present it before a judge or jury. This process could take anywhere from several months up to a year or more.
If there are multiple parties involved in the suit (for example: if one party accuses another of copyright infringement), then those court proceedings can take even longer because now there are more people involved who all have their own lawyers and need additional time to prepare their cases for trial as well.