The filing itself is quick once a policy is bound, because it is an electronic transmission from the insurer to the state. The time is spent placing the policy, which depends on how complete your information is when you call.
More detail
The bottleneck is almost never the state. It is the underwriting: getting a carrier to accept the record, confirming the vehicle and garaging details, and settling the down payment. A driver who calls with the DMV notice, the citation, an accurate vehicle list, and dates for any coverage gap can usually get quotes in one conversation. A driver who has to go looking for those documents mid-call adds days. Two things that genuinely slow it down: an unclear answer about what vehicle the driver has regular access to, which changes the form of the policy, and a gap in coverage whose dates nobody can pin down, which changes the price. Have both settled before you call and the rest moves at the speed of paperwork.