THE ANSWER is “instantly.” The brain is able to detect amplified sounds immediately after the insertion of a hearing aid, as long as the damage is not too significant.
Normally when sound enters the ear, acoustic information is relayed from the ear to the brain via nerve cells, called neurons. As the sound gets louder, more neurons fire simultaneously, which in turn allows the brain to detect the change in volume.

A hearing aid acts as a microphone, magnifying sounds that enter the ear. Hearing aids are mostly used in people who suffer from hearing loss because of damage to hair cells, the small sensory cells in the inner ear. Healthy hair cells can detect the magnified sounds from a hearing aid and convert them into neural signals. But the greater the damage to a person’s hair cells, the more severe the hearing loss and the more the hearing aid will need to make up the difference.

Hearing aids are able to help millions of people decipher sounds they could not access before, but these devices do not help everyone to the same degree. That is because although hearing aids make sounds louder, they do not repair or compensate for the damage that has taken place in the ear and the brain. As a result, hearing aids help the signals reach the brain, but the brain may not be able to process the signals, making the hearing aid less effective.

Source: Kelly Tremblay, associate professor of speech and hearing sciences at the University of Washington