By default, they'll probably rebook you onto flight(s) that will get you there quickest but that leave your origin at or after the original departure time. That is, they won't move you onto an earlier flight, unless you ask. You probably won't be given an overnight connection (and if you are, a hotel won't be paid, since the overnight will be part of the itinerary instead of due to a misconnection), so it's likely that they'll rebook the whole thing onto the first flights out the next day.

Your best bet is to find the new itinerary that works best for you, then call in and ask them to put you on it. Should be no charge. You'll probably be limited to just the airlines used in your original itinerary, but codeshares are probably allowed, and it doesn't hurt to ask about other *A flights.