Airport & Airway (Or ATC) slots are nothing to do with each other. When an airline is planning a route, from say Heathrow, to somewhere else the main thing to obtain will be the Airport slot. This is carried out through an organisation called ACL (Airport Coordination Limited) who carry this out on behalf of many airports in the UK.
Once slots at both ends are sorted out, if needed for the other end, if it is a busy airport, the airline will be free to fly that schedule for that particular season.
I am guessing the schedules from Airlines go to ATC so they know what levels of traffic to expect for a particular period. But 'slots' for airways are not issued in advance as a scheduled flight does not always follow the same route day in day out. Between LHR and ATH there could be 10 or so different routes available to an Airline. It will use whichever ones on a tactical basis due to winds, industrail action and other problems over particular sections of airspace.
ATC slots are then issued tacticaly on the day depending on traffic levels for a particular route. ATC slots are not issued for all flights, it is only those that pass through a section of airspace that has a restriction imposed on it for whatever reason, or for the destination airport that may have an arrival flow restriction, for example where LHR has restrictions due to Fog or High Winds.