Hoping to help.....As far as I know (Certainly at mine) major UK airports report runway surface conditions after an inspection by a member of Airfield Operations. The inspection is done on any change to weather conditions. A runway may be called wet from dry when it is obvious i.e in a heavy rain shower. but cannot be called dry from wet without the inspection.
ATC or an A/C Commander may request a runway inspection should there be any doubt. Damp is only called on discolouration of the service due to moisture. We have a grooved runway and call the runway wet if any moisture is present even in the grooves.
Some days it is hard to decide between damp and wet. On those occasions we would err on the side of caution and call wet.
We usually get to call the change from dry to wet within minutes and the ATIS is changed when we inform ATC. Changes from wet to dry take a bit longer on occasions as the runway can dry fairly rapidly in some instances. i.e a good wind. Once the rain stops we do try to inspect regularly to try and catch it ASAP,
It's a bit of a pain when a passing shower gives us a Wet, Damp, Dry call or Damp, Damp, Wet or similar oddities.
We do try to be as helpful as possible and will respond to direct requests through ATC for an inspection.
Any comments from Pilots would be welcome to ensure we get it right as often as possible.