B777-200 CREW ACCEPTED CLRNC FOR ANOTHER ACFT FOR DSCNT. AFTER THE APCH CTLR TOLD THE CREW TO MAINTAIN THEIR ALT; THE CREW HAD A TCASII RA WHEN THEY OVERSHOT THEIR ALT BY A RPTED 100 FT.

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I DID NOT HEAR THE LONDON DIRECTOR'S (APCH CTL) INITIAL XMISSION TELL US TO DSND TO 5000 FT; BUT THE CAPT RESPONDED TO THEIR XMISSION TO 'DSND; MAINTAIN 5000 FT.' THE RADIO XMISSION WAS APPARENTLY BLOCKED. FO WAS FLYING MANUALLY; AND HE STARTED TO LEAVE FL080. IN THE MEANTIME; ATC CAME BACK TO SAY THAT THE 2 ACFT RESPONDED TO THE LAST XMISSION; AND SAID THING LIKE 'THAT'S FOR ACR Y; DSND AND MAINTAIN 5000 FT.' THERE WAS A MOMENT OF CONFUSION TO WHO SAID WHAT AND WHAT EXACTLY ATC WANTED EACH ACFT TO DO. AT THAT TIME; WE WERE ALREADY APCHING FL078. CAPT RESPONDED TO ATC AGAIN THAT 'ACR X; DSND MAINTAIN 5000 FT.' SEVERAL RADIO XMISSIONS WERE BLOCKED. ATC EVENTUALLY CAME BACK AND SAID; 'NEGATIVE ACR X; THAT IS FOR ACR Y; NOT EVEN CLOSE. ACR X; MAINTAIN FL080.' FO IMMEDIATELY STARTED TO CLB BACK TO FL080 FROM FL078. THERE WAS AN ACFT (NOT ACR Y) XING R TO L IN FRONT OF US 1000 FT ABOVE. SINCE THE CLB RATE WAS SOMEWHERE AROUND 500-800 FPM; AND FO SLIGHTLY PASSED FL080 AND REACHING FL081. TCASII RA TRIGGERED. FO IMMEDIATELY LEVELED AROUND FL081; THEN DSNDED TO FL080. WE HAD CONFLICTING ACFT IN SIGHT ALL THE TIME; WAY ABOVE US AND THERE WAS ABSOLUTELY NO THREAT OF COLLISION. THE RA LASTED ONLY A FEW SECONDS. THE CHAIN OF EVENTS HAPPENED VERY FAST. IT STARTED WITH SEVERAL BLOCKED XMISSIONS WITH THE ATC AND LED TO THE TCASII RA. AFTER THAT; WE MADE AN UNEVENTFUL NORMAL LNDG TO RWY 9L IN LHR.