Processing time for DBA
It depends on the state and many times county or even town where your business is located (if its sole proprietorship then the state/county/town where you live).
Typically in many states you can visit the county/town clerk and obtain a DBA registration on the spot. Sometimes it would kill half a day, but you would have it done same day. Most of them require the application to be notarized (but provide notary public services on premises) and most of them require a money order as a form of payment (not cash).
If mailed in it could be anywhere between few days and few weeks, again, depending on the state/county/town.