Most would say the letters (So N, S, W, R, L), but also NSW Rugby League or the full New South Wales Rugby League would be used.
However, in most cases these competitions were called by the competition name (Winfield Cup, Reserve Grade, Jersey Flegg) or First Grade/Second Grade etc. rather than by the organisation which would only be used when referring to it specifically rather than the competition.
PSA: May be some delays with NAFL over the next week or so, currently moving into a new place and internet isn’t connected yet so fun times! Will try to get a round up in the next couple of days though in between all the unpacking etc.