A319 FLT CREW SQUAWKS 7700 DURING AN EMERGENCY DEVIATION FOR LEVEL FIVE THUNDERSTORMS ON DEPARTURE.

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BECAUSE OF SEVERAL LARGE TSTMS; ATC NEVER GOT US ON THE DEP PROC. INSTEAD WE WERE TOLD TO INTERCEPT THE ZZZ 084 DEG RADIAL. WE WERE GIVEN 3 DIFFERENT VECTORS AS THE CTLR COULD NOT FIGURE OUT THAT THERE WERE STRONG WINDS OUT OF THE S. 3 DIFFERENT TIMES THE CTLR TOLD US TO ACCELERATE TO 280 KTS AND 3 TIMES WE TOLD HIM THAT WE WERE GOING TO FLY 250 KTS FOR THE VERY BUMPY CONDITIONS IN THE CLB. THE FREQ WAS BUSY AND HE GAVE US DIRECT TO VOR. IT WORKED FOR THE TIME BEING SO WE TOOK IT BUT WX WAS GOING TO BE A FACTOR. WE LEVELED AT FL290 AND WERE FACED WITH A LEVEL 5 TSTM (LARGE RED AND PURPLE AREAS EXTENDING FOR MI). WE NEEDED A TURN TO THE S OF ABOUT 20 DEGS AND TRIED TO GET THE CTLR TO ACKNOWLEDGE 3 TIMES. HE FINALLY ANSWERED AND SAID NO TURN AND THE CAPT CAME ON THE RADIO AND SAID THAT HE WAS TURNING 20 DEGS R FOR THE TSTM. THE CTLR SAID TO MAKE IT A 'SLOW' TURN. 10 SECONDS LATER THE CAPT SAID HE WAS ESTABLISHED AND STATED THE HDG. THE RADIOS WERE VERY BUSY AND WE WERE PASSED TO ANOTHER CTLR ON THAT HDG. PEOPLE WERE STEPPING ON EACH OTHER AND SOMETIMES THE CTLR WOULD JUST GO SILENT. WE WERE AWARE OF THE ACFT AROUND US ON TCAS AND ANOTHER TSTM WAS ON OUR NOSE. WE HAD AN OPTION TO SHOOT A GAP ON THE L OR MAKE A HARD 60 DEG TURN TO THE R TO AVOID IT. AT LEAST 3 ATTEMPTS WERE MADE TO GET CTR TO ANSWER US AND FAILING TO COM AND WITH A SEVERE TSTM LOOMING; THE CAPT DIRECTED THE FO TO SQUAWK 7700 AND MADE A 60 DEG TURN TO THE R TO KEEP FROM ENTERING THE TSTM. THE RADIO GOT BUSY WITH OTHERS TRYING TO DEVIATE AND AFTER A MIN OR TWO THE CTLR ASKED IF WE WERE AN EMER. THE CAPT STATED THAT HE WAS IN AN EMER DEV OF 60 DEGS TO KEEP FROM GOING INTO A LEVEL 5 STORM AND STATED HIS HDG. THE CTLR SAID THAT THAT WAS AN OK HDG AND TO CONTINUE. THE CAPT STATED THAT AS SOON AS THE CTLR COULD TAKE POSITIVE RADAR CTL AND FLT CLRNC BACK; THAT WE WERE READY TO SQUAWK NORMAL. THE CTLR GAVE US OUR ORIGINAL SQUAWK BACK AND THE REST OF THE DEV WAS UNEVENTFUL. SUPPLEMENTAL INFO FROM ACN 787627: WITH SEVERAL WX CELLS BUILDING TO THE E OF ZZZ; WE WERE GIVEN ALTERNATE DEP INSTRUCTIONS SHORTLY AFTER TAKING OFF. WE WERE GIVEN A HDG TO INTERCEPT THE 084 DEG RADIAL FROM ZZZ VOR FOR WX AVOIDANCE. THE RIDE WAS ROUGH SO WE KEPT OUR SPD AT 250 KTS ON THE CLBOUT. ATC ASKED US TO MAINTAIN 280 KTS OR GREATER; WHICH WE TOLD THEM WE COULD ONLY DO 250 KTS UNTIL THE RIDE IMPROVED. THE SAME CTLR ACKNOWLEDGED AND STILL REPEATED THE SAME INSTRUCTIONS TO US 2 MORE TIMES WITHIN A FEW MINS. FINALLY; ATC RELENTED AND ISSUED US 250 KTS OR GREATER AND ASKED FOR 280 KTS OR GREATER WHEN ABLE; AND TO ADVISE THEM WHEN WE WERE INCREASING OUR SPD. A FEW MINS LATER; WE WERE ESTABLISHED ON THE 084 DEG RADIAL AND LEVEL AT FL290. THIS RADIAL WHICH WAS GIVEN FOR WX AVOIDANCE WAS NO LONGER A GOOD TRACK THROUGH THE WX AND WE REQUESTED TO DEVIATE 15 DEGS TO THE R. AT FIRST WE HAD A HARD TIME GETTING A WORD IN ON THE CONGESTED FREQ. WHEN WE DID STATE OUR REQUEST; IT WAS IGNORED -- NO RESPONSE. AFTER A COUPLE MORE CALLS; WE FINALLY GOT OUR REQUEST ACKNOWLEDGED AND BEGAN OUR DEV. THE NEXT FREQ WAS BEING OPERATED BY ANOTHER CTLR WHO ALSO SEEMED TO BE A BIT OVERWHELMED. WE BEGAN TO FEEL THAT THERE WAS ATC TRAINING GOING ON. NUMEROUS ACFT WERE REQUESTING DEVS WITHOUT ATC RESPONSE. WE WERE FAST APCHING ANOTHER CELL WE NEEDED TO DEVIATE AROUND AND GOT NO RESPONSE FROM OUR REQUESTS. THIS IS WHEN THE CAPT TOLD ME TO SQUAWK 7700 TO GET THEIR ATTN BECAUSE WE NEEDED TO DEVIATE IMMEDIATELY AND NEEDED TO MAKE OUR INTENTIONS KNOWN. WITHIN ABOUT 1 MIN; ATC CALLED US TO ASK IF WE WERE DECLARING AN EMER. THE CAPT STATED TO ATC THAT WE WERE DEVIATING; PRESENTLY ON A HDG OF 150 DEGS; AND IF THE CTLR HAD US IN RADAR CONTACT AND COULD PROVIDE ASSISTANCE; WE WOULD RETURN TO OUR PREVIOUS XPONDER CODE. OTHERWISE; THE CAPT WAS GOING TO EXERCISE EMER AUTH TO MAINTAIN SAFETY OF THE FLT. AT THAT POINT; WE WERE TOLD WE WERE IN RADAR CONTACT; AND TO SQUAWK OUR PRIOR CODE; AND DEVS APPROVED. WE FINISHED OUR DEVIATING; RETURNED TO OUR RTE AS INSTRUCTED BY ATC; AND COMPLETED OUR FLT WITHOUT ANY FURTHER ISSUES. WE DON'T KNOW IF IN FACT THERE WAS CTLR TRAINING GOING ON; BUT 2 OF THE CTLRS WE DEALT WITH DEFINITELY SEEMED TO BE OVERWHELMED AT THAT TIME.