N90 Controller receiving OJT on combined positions described a loss of separation event when FOK RNAV Runway 24 approach traffic executed a missed procedure; conflicting with an enroute IFR SR22; both student and instructor failed to noted the missed approach altitude resulting in the mishap.

Narrative

I was receiving OJT (on job training) on the South Side combined (ISP; CCC; BEADS HI; BEADS LO; BEADS HAND OFF; COORDINATOR). The weather was IFR. Aircraft X was on an IFR flight plan eastbound level at 3000 FT direct BIGGA WP vectored for the RNAV Runway 28 approach at HTO. Aircraft Y was also on an IFR flight plan inbound for FOK from the northeast. Aircraft Y was asked if he had the current FOK weather and to state intentions. Aircraft Y stated they had the current weather and wanted the ILS Runway 24. I advised Aircraft Y that Runway 24 has been NOTAM closed for over a month and asked again to say intentions. Aircraft Y was also advised that Tower was reporting ceiling was OVC 003 and again asked to say intentions. Aircraft Y said that they need some time to figure out what they want to do. After some time Aircraft Y came on frequency and requested the RNAV Runway 24 circle to land Runway 19. I coordinated with FOK Tower and vectored Aircraft Y for the approach as requested. Once Aircraft Y was cleared and established on the final approach course I terminated RADAR service and instructed Aircraft Y to contact FOK on the appropriate frequency. After a few minutes I began to feel that something was not right at which point I looked in the IDS 4 at the missed approach for the FOK RNAV Runway 24 and realized the miss climbs to 2600 southwest bound direct ZATBO WP to hold. The IFR Aircraft X was in the area of ZATBO WP eastbound level at 3000 FT as previously stated. I realized the altitude of the FOK RNAV Runway 24 missed approach (2600) was a conflict. At that moment I observed Aircraft Y in a climb out from FOK. I instructed the Aircraft X to turn right heading 150 no delay to which he responded to. Next transmission was for the Aircraft X to climb and maintain 4000 no delay. I did not get a reply this time. FOK Tower informed me that Aircraft Y was on the go. Once Aircraft Y checked on he was already out of 2200 and instructed to descend to 2000 no delay and to identify. After Aircraft Y was RADAR identified he was given a right turn heading 360. Aircraft X was instructed to climb and maintain 4000 again to which he complied. Separation was reestablished and all parties involved continued on there ways. Recommendation; I will be refreshing my knowledge of the missed approach charts for the airports in my sector to ensure this will not happen again.