The whole purpose of designating a takeoff alternate on the flight plan, if required by the conditions at the departure airport, is to 1) make sure you have a suitable diversion field available (i.e. check wx, notams, etc) in case of airborne emergency immediately after takeoff, and 2) provide ATC that information in case of a radio out situation so they know your intentions. Once you are airborne you are free to change up the plan in coordination with ATC.