A300 WITH ZAB IS GIVEN CLB FROM FL330 TO FL370 AND CONFLICTS WITH ACR.

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ZAB CLRED US TO RWY 37 FROM RWY 33. AT FL353 ABQ ASKED WHEN WE WOULD BE LEVEL AT FL370. I INFORMED THE CTR IT WOULD BE 2 MINS. WE WERE CLBING AT 1000 FPM; AND OUR COMPUTER SHOWED US LEVEL AT THAT TIME. AT FL360 THE WIND SHIFTED AND OUTSIDE AIR TEMP STARTED TO INCREASE; WHICH SLOWED OUR RATE OF CLB. OUR COMPUTER STILL SHOWED US LEVEL ON TIME; WITH 1+ MIN TO GO. AT FL365 ATC ASKED US IF WE WERE GOING TO MAKE FL370 AND WE NEEDED TO BE THERE IN 1 MIN. AGAIN I STATED WE WOULD BE THERE. I ASKED ATC IF THEY WANTED US TO TURN THAT WE HAD NO PROB WITH THAT. ATC TURNED US 40 DEGS AND ACR Y 40 DEGS. AGAIN ATC ASKED IF WE WERE LEVEL AT FL370; WE WERE PASSING 36850 FT WITH 20 SECONDS STILL TO GO ON OUR TIME TO BE LEVEL. ABOUT 2 MINS LATER ATC ADVISED US THAT WE HAD A PLTDEV AND TO CALL THEM. (#1; WE MADE FL370 IN TIME. #2; ATC NEVER CALLED TFC. #3; ATC FAILED TO HAND US OFF. #4; AFTER CALLING ZAB THE SUPVR STATED IT WAS TOO CLOSE AND WE SHOULD HAVE CALLED LEVEL AT FL370.) ABQ SAID THE RADAR SHOWED US 4.8 MI AND FL365; BUT OUR ACFT WAS AT FL370 BEFORE ACR Y PASSED.