Depends on aircraft type.
In Beaufighter or Mosquito, for example, the radio operator (Wireless Op in RAF parlance) would also be the navigator and, if fitted, radar operator /observer.
In bombers, such as the B-17 or Lancaster, he would normally be radio op only.(or Wop/AG - Wirelss Op /Air Guner - in some RAF aircraft.)
Radios of the time were big, bulky affairs, with separate, valve-powered receiver and transmitter, and often more equipment such as antenna tuners, trimmers, tail radar warning etc etc., normally operating in the HF band.
He would receive info on current winds at varying altitudes, as well as other Met info, and would be able to get a navigational 'fix' for the navigator by holding down his Morse key for a certain length of time, to transmit a given signal, when the return signal from a ground station would provide Lat and Long.
The radios operated on W/T, and Morse messages were mostly sent in code. He would also be able to control the frequencies of the R/T, so that the pilot could talk in voice to other aircraft or ground stations on the same frequencies, normally operating in the VHF band.
He would also receive any information concerning diversion of target(s), change in conditions over the target, or any recall order, and so on.
It was (and is) a quite complex and important role, and the qualified operator would not only have to be able to send and receive Morse at a high rate, but he would also need to be able to calculate antenna lengths for a given frequency at varied altitudes and ranges in differing atmospheric conditions, at differing times of the day or night. Apart from the fixed wire antenna on WW2 aircraft, most large types also had a 'trailing' aerial, a long wire on a reel, with a weight on the free end, which could be reeled out to the required length for best signal reception.
As the war progressed, the role became more complex, and in some instances the radio op would also be responsible for some radar equipment, including such items as 'Fishpond', a rear-warning radar, and assist the navigator with up-dates on wind, position etc etc.