How long does it take to purchase a tax deed property?

No idea; not my area of expertise.

This may be better asked in real estate.

It takes about 3 minutes, the property is sold once a year at the court house steps at auction, the price starts at the amount of tax due on it.
Sometimes the price will go very high, other times it sells for just the opening bid.

Each state is different, in many states you get a tax deed the day of the sale after you pay the price in full, normally in cash unless prior arrangements made.

In some other states you merely buy a tax certificate which you exchange for the deed in a specific amount of time. This time varies from one year to three years. During this time, you do not own the property, just own the tax lien, so you have no rights to the proeprty. And the property owner has the right to redeem his property for the amount you paid plus a set interest ( normally about 10 percent)