Going to court, or settling out of court, for an asbestos lawsuit can take any amount of time, depending on the specific details of the case. In many situations, you won't be able to tell how long your case will take until you know whether or not the other side is willing to settle, or if you will have to go to trial. Obviously a trial will take longer than a settlement would.
In general, the first several steps of an asbestos lawsuit will involve getting a lawyer and providing them the information they require, as well as the filing of the lawsuit itself. If your lawyer is available and you are ready to move on this, you could finish this initial process within a week or two. Once the papers are filed, it will be up to the other side to respond. You may want to give them a deadline, such as ten days, so that they don't drag their feet too long in responding.
Since every asbestos lawsuit is different, this means there is no set timeframe. Your best option is to contact a lawyer who can evaluate your particular case and give you insight on how to proceed and how long the case may take.