There really is no easy answer to that question.
If it's a clear and simple matter such as unpaid taxes, the process moves very quickly. If fact, the title underwriter may ask the title agent to pay the taxes and then follow up with a reimbursement check to the agent.
Other types of claims will take longer because first there is a determination of the claim validity, then a determination of the amount of damages.
From the perspective of the insured, it's really important to contact the title company as soon as you think there may be a claim. Having the title company involved from the beginning may mitigate damages and make the resolution move more quickly.