More than likely the authorities wouldn't tell anyone it was a dirty bomb anyway for at least a week out of the correct assumption that releasing the information would cause more panic in the populous and make the situation even more difficult for the authorities in the clean up. All these publically held exercises are essentially just public relation exercises to show the public that they are prepared for the worst.

You will have to rely on your own instincts, if there was an incident i.e. large explosion and you get a metallic taste in your mouth (nuclear radiation fallout is notoriously difficult to detect from your own senses) it might be prudent to do you own Decom. i.e. get yourself booked into a good well built hotel (most hotel bathrooms are internal with no outside windows) after purchasing some new clothes for a good shower, scrub down and mouth wash as quickly as possible. If you don't have time to purchase clothes at least you will have a hotel bathroom robe to wear.