Here are the umpire mechanics. They are based on the concept that a runner who passes a base is always assumed to have touched it, until/unless the missed base is properly appealed.
1) (Missed first base) The umpire should call and signal "safe". The defense may appeal the missed base, but it must be appealed before the runner returns to and touches first base (either colored base). If properly appealed, then the runner is out. If not, then still safe.
2) (Missed home plate on force play) Treat the same as play at first base.
3) (Missed tag of runner/non-force play) The umpire should briefly pause to see if either player will immediately try to complete the play (either runner scrambling back to touch home or fielder attempting to re-tag), then call and signal "safe". The defense may appeal the missed base, but it must be done before the runner retouches the plate. If the runner leaves the field with no appeal, then she is still safe, but has lost her right to come back and retouch. She would still be subject to an appeal and would be out if an appeal is made (before the next pitch).