CAPT OF ACR WDB ACFT RESPONDED SLOW TO AN ATC REQUEST TO CLB RESULTING IN A CONFLICT.

Narrative

SCHEDULED PAX FLT STR-AMS WAS HELD TO FL240 BY FRA CTR. ACFT WAS ON AUTOFLT WITH THRUST MGMNT SYS IN OP. CTLR ASKED US TO CLB TO FL250. CAPT RESPONDED TO HIM; 'ROGER; CLBING 250.' CTLR THEN CAME BACK AND ASKED US TO EXPEDITE CLB. FO WAS FLYING ACFT; HAD PROGRAMMED INTO AUTOFLT SYS FOR ACFT TO MAKE AUTOFLT CLB; HE THEN WENT TO VERT SPD 1000 FT AS THAT WOULD BE A CONSIDERED MOSTLY MAX CLB JUST CLBING 1000 FT IN ORDER NOT TO OVERSHOOT 25000 FT. CTLR THEN CAME BACK AND ASKED; 'ARE WE EXPEDITING;' CAPT REPLIED; 'YES WE ARE.' ABOUT THAT TIME WE LOOKED OUT THE WINDOW ABOUT 1000 METERS R TO L A WHITE WDB ACFT WITH A MAROON TAIL. CAPT ASSUMED MANUAL CTL OF ACFT; PULLED BACK CTL WHEEL AND WE WENT OVER THE TOP OF THE ACFT. ALL 3 CREW MEMBERS SAW THIS. CREW MEMBERS ESTIMATE MAYBE 500- 600 FT; SOMETHING LIKE THAT. AGAIN WE WERE ASKED R AFTER THAT 'ARE WE LEVEL 250;' AND BY THEN WE WERE. CAPT ASKED CTLR WHAT ALT OF WDB WAS AND HE REPLIED FL240. NOTHING WAS SAID BY CREW AS WE DID NOT WANT TO MAKE AN ISSUE OF THIS OVER THE AIR. CTLR THEN IMMEDIATELY HANDED US OFF TO NEXT SECTOR. THIS WAS RPTED TO COMPANY FLT OPS UPON ARR AT AMS. WHAT COULD HAVE PREVENTED THIS? EVIDENTLY THE ENTIRE TIME TRANSPIRING FROM THE TIME OF THE FIRST CALL UNTIL THE ACFT WAS SIGHTED COULD NOT HAVE BEEN MUCH OVER 30-35 SECONDS. EVIDENTLY THE WDB ACFT WAS QUITE CLOSE TO BEGIN WITH WHEN THE CALL WAS MADE; OTHER THAN THAT I DON'T KNOW REALLY WHAT OTHER CIRCUMSTANCES WERE INVOLVED. I HAVE A FEELING; KNOWING WHAT GOES ON WITH THAT SECTOR; THAT ACFT WAS PROBABLY CTLED BY ANOTHER SECTOR ON AN INBOUND TO FRA. I KNOW THERE ARE SOME PROBLEMS WITH THIS AREA. CTLR SHOULD HAVE GIVEN US FL250 EARLIER; I'M SURE. WHEN HE SAID TO EXPEDITE TO 250; MY FIRST THOUGHTS WERE HE WANTED TO HAND US OVER; WHICH IS THE NORMAL SITUATION THERE. HAD I KNOWN THERE WAS A WDB ACFT; 5-6 MI FROM US AT THE TIME TO BEGIN WITH; THE NORMAL EXPEDITE USING AUTOFLT DEFINITELY WOULD NOT HAVE BEEN USED.