What can cause an itchy rash with small bumps 15 to 20 min after swimming in any swimming pool?

Dry skin. The chemicals in pool can strip your skin of its protective oils. Apply baby oil before and after swimming.
Do some observations. To find out what might be causing a rash after 15-20 minutes in any swimming pool, one needs to do some observations. If it's the chlorine, then swimming in a lake won't cause the rash. If it's the cold water, then using a pool of warmer water won't cause the rash. If the body is trying to sweat due to heat generated by swimming, then sitting calmly in the pool shouldn't cause the rash; etc....
Cold urticaria. There are several things to consider, such as allergy to the chlorine or salt (depending on the pool), dry skin, or cold urticaria. Cold urticaria is essentially hives when the skin gets cold. Test with an ice cube on the skin for a reaction. Talk with your doctor for other suggestions, since a wide variety of issues could cause this.