Actually, once the Mass starts, the priest’s part is not “done” until he imparts the final blessing. Even should he complete the Consecration by consuming the Sacred Species before he is unable to continue, another priest must finish for him. A deacon or lay person cannot come in following the priest’s Communion and finish… it must be a priest.
I seem to remember reading either on other threads here or at EWTN that should the priest become physically unable to complete Mass that the Head Usher takes command of the situation, orders the doors locked, and delegates someone to call emergency services, ushers to watch for the ambulance, and someone to contact another priest to complete the Mass. The congregation is technically not allowed to leave until another priest arrives to finish the Mass, which is why the doors are locked. Obviously the ushers watching for the ambulance and the new priest open the doors to admit emergency services personnel and the new priest.
Now, in the case of the OP, because the Final Blessing has been imparted, the Head Usher still takes charge to get help for the priest, but there is no need to keep the Faithful. The Mass has ended, if their help is not required, they should depart.
Once the Eucharistic Prayer is complete the Priest is no longer needed for it to be called a Mass. It has happened twice at my parish. Both time the priest left the alter because a family member ran into the church urging the priest to come and administer last rigthts and a final confession.
Both times our deacons were present and finished up with distributing communion and final blessing.
As I understand, this should not be occuring either. The priest’s primary responsibility is to complete the Mass he has started, not to annoint and hear final confession. The priest-celebrant must be the one to read the post Communion Prayer and to give the Final Blessing. The deacons do not have faculty to do these at Mass as they are texts reserved to the priest in the context of a Mass.