I've come accross this type of claim before, but not specifically with DCC.
I would assume that they'd look at whatever you sent in and cross reference it against internal information i.e. check that council was working at your location on date of damage and seek report/verions of events from whoever was in charge of the work. If not clear cut that it is or isnt their fault, they may send someone out to look at your property and/or talk to you.
After considering all of the above, they'll likely take a view on the incident and if it is there fault, make you an offer to settle the claim.
Timeframe for all of the above depends on how quickly they can establish the facts. Could take weeks or even months.