TRACON controller described a loss of separation event when multiple pilot requests for weather avoidance precluded a timely transmission required for separation.

Narrative

There was a strong cold front passing through the area causing very strong winds out of the west-southwest. Aircraft X was north-northwest of ZZZ and was being vectored for an ILS Runway 17L descending to 3;000 ft. Aircraft Y was east of ZZZ northbound being vectored for an ILS Runway 18R at 3;000 ft. I turned Aircraft Y to the northwest and the pilot questioned me about the weather on the Runway 18R final approach course because he didn't want to fly through it; so I turned Aircraft Y back to the north. I went on to turn Aircraft X onto the Runway 17L final approach course; but I immediately noticed my transmission had been blocked by the pilot of Aircraft Y; complaining about weather. As soon as the pilot of Aircraft Y finished talking I immediately turned Aircraft X to the southwest and told the pilot to expedite the turn. I climbed Aircraft Y and descended Aircraft X; all in an effort to prevent a loss of separation. The inability to make all my transmissions in time because the pilot of Aircraft Y was complaining about the weather caused me to lose separation between Aircraft X and Aircraft Y.