If a man is ordained by a SSPX consecrated bishop using the traditional Roman Catholic ordination rite, but the bishop uses "Holy oil" that was "consecrated" by a novus ordo-ordained-and-consecrated "bishop", will the ordinand's hands be consecrated?

He will be a priest but...will he be a new kind of hybrid priest?  What would be the remedy?

I answer that, As stated above (A. 4), to cause the form and to give the matter its proximate preparation for the form belong to the same agent. Therefore, the bishop in conferring holy orders does two things, for he prepares the candidates for the reception of holy orders and delivers to them the power of order.

He prepares them both by instructing them in their respective offices and by doing something to them, so that they may be adapted to receive the power. This preparation consists of three things: namely, blessing, imposition of hands, and anointing. By the blessing they are enlisted in the divine service; thus the blessing is given to all. By the imposition of hands the fullness of grace is given, whereby they are qualified for exalted duties; thus only deacons and priests receive the imposition of hands, because they are competent to dispense the sacraments, although the latter as principal dispensers, the former as ministers. But by the anointing they are consecrated for the purpose of handling the sacrament; thus the anointing is done to the priests alone who touch the body of Christ with their own hands, even as a chalice is anointed because it holds the blood, and the paten because it holds the body.

The conferring of power is effected by giving them something pertaining to their proper act. And since the principal act of a priest is to consecrate the body and blood of Christ, the priestly character is imprinted at the very giving of the chalice under the prescribed form of words.

Interesting thought.  Now, the entire man becomes the Persona Christi, and receives the character of Christ, so in that sense it may not be a sacrilege, but I'm on the fence here.

Reply Obj. 4: The bishop receives power to act on Christ’s behalf upon His mystical body, that is, upon the Church; but the priest receives no such power in his consecration, although he may have it by commission from the bishop. Consequently all such things as do not belong to the mystical body are not reserved to the bishop, such as the consecration of this sacrament. But it belongs to the bishop to deliver, not only to the people, but likewise to priests, such things as serve them in the fulfillment of their respective duties. And because the blessing of the chrism, and of the holy oil, and of the oil of the sick, and other consecrated things, such as altars, churches, vestments, and sacred vessels, makes such things fit for use in performing the sacraments which belong to the priestly duty, therefore such consecrations are reserved to the bishop as the head of the whole ecclesiastical order.

I answer that, As stated above (A. 1), the sin of sacrilege consists in the irreverent treatment of a sacred thing. Now reverence is due to a sacred thing by reason of its holiness: and consequently the species of sacrilege must needs be distinguished according to the different aspects of sanctity in the sacred things which are treated irreverently: for the greater the holiness ascribed to the sacred thing that is sinned against, the more grievous the sacrilege.

Now holiness is ascribed, not only to sacred persons, namely, those who are consecrated to the divine worship, but also to sacred places and to certain other sacred things. And the holiness of a place is directed to the holiness of man, who worships God in a holy place. For it is written (2 Macc 5:19): God did not choose the people for the place’s sake, but the place for the people’s sake. Hence sacrilege committed against a sacred person is a graver sin than that which is committed against a sacred place. Yet in either species there are various degrees of sacrilege, according to differences of sacred persons and places.

In like manner the third species of sacrilege, which is committed against other sacred things, has various degrees, according to the differences of sacred things. Among these the highest place belongs to the sacraments whereby man is sanctified: chief of which is the sacrament of the Eucharist, for it contains Christ Himself. Wherefore the sacrilege that is committed against this sacrament is the gravest of all. The second place, after the sacraments, belongs to the vessels consecrated for the administration of the sacraments; also sacred images, and the relics of the saints, wherein the very persons of the saints, so to speak, are reverenced and honored. After these come things connected with the apparel of the Church and its ministers; and those things, whether movable or immovable, that are deputed to the upkeep of the ministers. And whoever sins against any one of the aforesaid incurs the crime of sacrilege.

The practice of the early Church was, for centuries, to place Holy Communion into unconsecrated hands.  So unless we want to conclude that the Church endorsed sacrilege, we must conclude that, despite all the good reasons to have moved away from that practice, placing the host in unconsecrated hands is not, in se, sacrilege.

Your idea is an expression of what Pius XII called archeologism or "antiquarianism" (Mediator Dei, 64). Antiquarianism was the strategy of the reformers after Vatican II (following the strategy of the Protestants in the 1500s).

Traditio in Latin means "to hand over," to pass the baton. The baton (the Faith and the Sacraments) develops over the centuries based on reason and the Magisterium, without ever contradicting what was definitively taught before. Think of a stick (baton) that was originally rough but over centuries of being passed down becomes more smooth. We can't and should not want to go back and add onto the baton all the rough spots that were originally there. They were smoothed away for good reason.

The early Church had not had the chance to think clearly about the implications of unconsecrated hands touching the Eucharist. That came later. On this matter, the Church teaching and customs "developed" for good reasons. It would be unreasonable and sacrilegious to accept that early-Church practice understanding the Eucharist as we do now.

Here is what St. Thomas says (https://aquinas.cc/la/en/~ST.IIISup.Q37.A5):

So, if the "anointings" described above are not done with true Chrism (which requires the blessing of a traditionally-consecrated bishop), there would presumably be a sacrilege occurring whenever the priest or the vessels touch Our Lord.