There is no way to tell how long it will be until you are pain free.
Generally speaking, the more chronic your issue, the longer it will take to resolve and the more recent your injury the quicker it should resolve. Obviously this is all dependent on what the actual problem is.
Muscle strains can take as little as a few days resolve, whereas something like a disc prolapse may not respond to treatment at all (although the pain associated with it can be reduced). Ligament and tendon issues may take months to resolve.
Your osteopath will give you things to do in between sessions such as exercises, stretches, nutritional or lifestyle changes. The more you can do consistently, the better your outcome will be and the faster you’ll improve.