Good question. Originally the below snip applied, but I don't see how it can possible apply these days. Can you swear allegiance only as long as you're excluding everything anti-Catholic?

5:37 "...it is part of the Christian tradition that as the Roman Empire was dismantled, that throughout Europe the Christian Monarchies arose, and all of those Christian Kings recognized that they were to take Christ as their model.

They were to think of themselves as the vicegerents of Christ, just as the pope was the vicar of Christ upon the earth. And the people were taught they had two sovereigns on the earth, but both of them did represent Christ. The pope represented Christ in the spiritual realm, and the king represented Christ in matter temporal, and the people were to obey both as a religious duty..." - Fr. Wathen

I don't know how the oath is worded, but maybe a mental reservation could allow for it:  being truly loyal to them and their salvation and souls by not affirming them in evil and opposing them when they say/do things that put their souls in jeopardy.

This is a good question when one considers all the martyrs of the past who would never take such an oath.  

Depends entirely on what the oath said. St. Thomas More would have sworn allegiance to Henry VIII if he could have without acknowledging Henry's claim to be head of the English Church.

All these "Test Oaths" as they were called have been long repealed. The regular oath is this:

I, (full name), do swear that I will be faithful and bear true allegiance to His Majesty King Charles, his heirs and successors, according to law. So help me God.

For citizenship:

I... swear by Almighty God that, on becoming a British citizen, I will be faithful and bear true allegiance to His Majesty King Charles the Third, His Heirs and Successors according to law.

I don't see anything objectionable to that at all.

One could object that Charles is no king, but a usurper, just as Elizabeth was a "pretended Queen of England, and the servant of crime" according to Pope St. Pius V in Regnans in Excelsis.

Are Catholics obliged to consider heretics as illegitimate kings? Can a Catholic swear faithful allegiance to a heretic?

I wouldn't be surprised if the historical fact is that Catholics did swear allegiance to heretics.

Pope St. Pius V, Regnans in Excelsis:

[We declare Elizabeth] to be deprived of her pretended title to the aforesaid crown and of all lordship, dignity and privilege whatsoever.

And also (declare) the nobles, subjects and people of the said realm and all others who have in any way sworn oaths to her, to be forever absolved from such an oath and from any duty arising from lordship. fealty and obedience; and we do, by authority of these presents , so absolve them and so deprive the same Elizabeth of her pretended title to the crown and all other the above said matters. We charge and command all and singular the nobles, subjects, peoples and others afore said that they do not dare obey her orders, mandates and laws. Those who shall act to the contrary we include in the like sentence of excommunication.

St. Pius V excommunicated those who would consider her to be the Queen.  Therefore the line of British "royalty" is copmletely illegitimate.  Given the reasons for his commands related to Elizabeth, one would gather that St. Pius V would also disapprove of any who would pretend that the successors of Elizabeth were legitimate also.  If she ceased to be queen, then here successors are not legitimate royalty either.  St. Pius V excommunicated anyone who would treat her as having any authority (i.e. by obeying her orders).

Ironically, Elizabeth was no queen even before this, but had usurped the line, which now resides in the son of a truck driver from Australia.