I have just updated by parish website's Downloads Page to include a leaflet produced by the Guild of Catholic Doctors. It includes prayers for the dying and prayers to be said immediately after someone has died.
The prayers should be led by the priest if he is present - he would also give the sacrament of anointing, viaticum, and the apostolic blessing with the plenary indulgence. However, the prayers in this leaflet can be very helpful for the family and for the dying person to repeat again after the priest has gone, or if a priest cannot be had.
The family may request that the priest use the older form of the rites. However if the priest is unable to do so (because he doesn't have the relevant book, because he does not know any Latin etc.) it is obviously more important for the person to receive the sacraments than to worry about which rite is used.