How do we sweat immediately after drinking water on a hot day?
Not everyone sweats in seconds after drinking water in summer.

Also, sweating in seconds after drinking water is not likely due to water being absorbed (Nutrition Reviews, 2015):


  Ingested fluids are not immediately available for assimilation into
  the body. They are initially stored in the stomach, and there is
  little net absorption of water or solute across the gastric mucosa.


There are a lot of receptors, including stretch receptors and various chemoreceptors, in the stomach, so your stomach will likely recognize water as water and signal the brain via the nervous system, probably via the visceral afferent fibers.

One of two things may then happen:


Your brain can perceive a sudden appearance of water in your stomach as stress and react with sweating, like when you are embarrassed or excited.
When dehydrated, you sweat less to prevent further dehydration; when your brain recognizes there is new water in your stomach, it can now allow the signals coming from warm skin to trigger sympathetic nerves to release some sweat to cool you down.