The decision making process can vary for each museum and will take different lengths of time.
The transfer of ownership from you to the museum does not take place immediately you hand over the object. The person on the front desk usually does not have the responsibility to commit the museum’s time and money to caring for new objects.
Each potential new object is first considered by experts who consider how it fits with the collecting policy, what the costs of acquiring the object are, and what the objects needs are e.g. How much room will it take in the store? Will security have to be enhanced? They may have to research the object in order to see if it fits the collecting policy.
In some museums, the expert staff have the power to decide whether or not to accept an offer, in other museums, they can only recommend to a board of trustees who will make the decision.