A B737-800 FLT CREW RPTED THAT FOLLOWING A BLEEDS OFF TKOF; THEY FAILED TO COMPLETE THIER CHKLISTS AND TURN ACFT BLEEDS ON. CREW CITED WX DISTRACTIONS AND A DELAYED RECOGNITION OF THE WARNING HORN AS A CABIN ALTITUDE ALERT.

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AFTER CONDUCTING AN ENG BLEEDS OFF TKOF (PACKS ON APU; ENG BLEEDS OFF) I (PF) CALLED FOR THE AFTER TKOF CHKLIST. THE PM REPLIED; 'AFTER TKOF CHKLIST COMPLETE.' WX WAS GUSTY WITH MODERATE TURB. WE STARTED DISCUSSING THE CLB INTO THE CHOP; WITH OMINOUS LOOKING CLOUDS; AND KEEPING THE FLT ATTENDANTS SEATED; AND THE USE OF CTL WHEEL STEERING FEATURE OF THE AUTOPLT. WE BECAME ENGROSSED WITH THE WX AND THE RELATED FACTORS: AT APPROX FL210 TO FL220; THE CABIN ALT WARNING HORN SOUNDED. WE DONNED THE OXYGEN MASKS. THE PM STARTED TO RECONFIGURE THE PRESSURIZATION AND REGAIN CTL OF THE CABIN PRESSURE. THE PF LEVELED THE ACFT AT ABOUT FL240. WE COORDINATED A DSCNT TO FL200. BY THE TIME WE LEVELED OFF AT FL200; WE RECOVERED CTL OF THE CABIN PRESSURE. WE REQUESTED A CONTINUED CLB TO OUR FILED ALT. 1) ALTHOUGH THE PM HAD THE BLEED OFF CHKLIST OUT AND OPENED; WE ALLOWED OURSELVES TO BECOME DISTRACTED BY THE WX. WE NEVER FINISHED THE ADDITIONAL ITEMS OF THE AFTER TKOF CHKLIST LISTED IN THE BLEEDS OFF TKOF. THE PACKS HAD BEEN LEFT ON THE APU BLEED WHICH WAS NOT CAPABLE OF MAINTAINING CABIN PRESSURE. 2) WE ALLOWED THE EXISTING WX CONDITIONS TO BREAK OUR ROUTINE AND CREATE A NEW THREAT. THREAT MANAGEMENT REQUIRES THAT WE NOT BECOME MYOPIC TO THE POINT OF CREATING NEW OR INCREASED THREATS.CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THAT HE AND THE FO TOOK ABOUT 30 SECONDS TO IDENTIFY THE WARNING HORN'S CAUSE. THIS IS THE FIRST TIME THE RPTR HAS HEARD THIS HORN IN OVER 20 YRS OF FLYING. THE RPTR STATED THAT REALIZING THE CAUSE; THEY WERE RELUCTANT TO DON OXYGEN MASKS; BUT BOTH DID ANYWAY. AN EMER WAS NOT DECLARED; BUT THE ACFT WAS LEVELED UNTIL THE CABIN BEGAN DSNDING. THE RPTR WAS SURPRISED AT HOW LONG IT TOOK FOR THE CABIN TO BEGIN DSNDING. HE WOULD SUGGEST AN AURAL WARNING SPECIFYING CABIN ALT OR AT LEAST AN AIR CONDITIONING ANNUNCIATION ON THE FWD PANEL MASTER CAUTION LIGHT. DONNING THEIR OXYGEN MASKS DID KNOCK BOTH OF THEIR HEADSETS OFF; BUT BECAUSE THE CAPT HAD HIS SPEAKER ON; COMMUNICATION WAS QUICKLY REESTABLISHED.