Depending on the size of the addition, the process could take several months to a year.
It REALLY depends on a lot of factors. You would have to interview and select an architect and contractor, determine what approval process your local municipality requires, see how long the design and permitting phase usually takes, etc.
BUT there are some very important questions you would need to ask first:
Is the house on septic or sewer? This would affect your ability to add on, especially a bathroom.
What are the building setbacks on that lot? You may not be able to add on a particular side of the house.
Is there any flood plain on the lot? That can affect whether the building footprint can be expanded.
Have your agent ask these questions at the local building codes/planning office BEFORE you make an offer.
Also: Adding square footage will trigger a reappraisal and will increase the amount of property tax you would pay.