B757 OFF RWY 29 EWR ON THE EWR 6 SID CLBING TO 5000 FT. MODERATE TURB IN THE CLB AND PROB CONTACTING DEP CTLR. OUT OF 4000 FT HAD TCASII TA FOLLOWED BY RA AT 4600 FT. FOLLOWED ALERT AND GOT TO 5400 FT TO CANCEL THE RA. ON CONTACT WITH CTLR ADVISED TO CHANGE XPONDER CODE AND CLRED TO 6000 FT.

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ON DEP FROM RWY 29 AT EWR IN WHICH THERE WERE SOME VERY STRONG SURFACE WINDS; IE; 280/23G42 ON DEP THE COPLT WAS FLYING AND WE WERE TOLD TO FLY THE EWR 6 DEP AS PUBLISHED WHICH IS RWY HEADING TO THE MARYANN FAN MARKER; THEN FLY HDG 265 DEGS AND CLB TO 5000 FT. THE CLBOUT WAS EXTREMELY CHOPPY WITH A TURB LEVEL OF BTWN CONTINUOUS TO FREQUENT MODERATE TURB. AIRSPD LOSS OF PLUS OR MINUS 10 TO 15 KTS. TWR ADVISED US TO CONTACT DEP ON 119.2; I DID CONTACT THEM AND RECEIVED NO REPLY. AT ABOUT THE SAME TIME WE RECEIVED A TCASII TFC ALERT; ON A HDG OF 265 DEGS AND CLBING THROUGH 4000 FT. I IMMEDIATELY SELECTED THE 5 MI SCALE AND NOTICED THE TARGET AT 12 O'CLOCK AND ABOUT 2 1/2 MI; THE TA TURNED INTO AN RA WITH A COMMAND TO CLB AND MONITOR VERT SPD. OUR CURRENT ALT WAS BTWN 4600-4700 FT MSL. WHEN THE RA WAS ALERTED WE FOLLOWED THE RA UNTIL THE CONFLICT WAS NO LONGER A THREAT. I THEN RADIOED NY DEP CTL TO TELL THEM WE HAD AN RA AND WE WERE RETURNING TO OUR ASSIGNED ALT OF 5000 FT MSL. THE ALT NEVER WENT ABOVE 5400 FT MSL TO SATISFY THE TCASII ALERT. ONCE COMS WAS ESTABLISHED WITH DEP CTL THE CTLR WAS ASKING WHO WAS THE TFC W OF EWR WHEN WE TOLD HIM IT WAS US AND HE THEN ADVISED TO SQUAWK XXXX WHEN XXYZ WAS IN THE BOX. WE EVENTUALLY WERE TOLD TO CLB 6000 FT MSL AND THE EVENT ENDED. IN RETROSPECT I THINK THE RAPID CHANGE IN WX CONDITIONS WERE OF GREAT CONCERN TO US AND ALSO THE ATC FOLKS.