I don't remember there being any actual figures for snow depth etc. set out at which an airfield must be closed. In the U.K. it is the Airport Management - not ATC - who have responsibility for making this decision. It can be a very difficult decision.
The airport personnel have to inform ATC of the "runway state" and ATC are responsible for passing this information to pilots. As always, the final decision as to whether to operate rests with the pilot in command.