I read that he was the favourite nephew of his uncle Charles of Lorraine so he likely also inherited from him (Charles of Lorraine left a fortune to an illegitimate son).
Since Charles of Lorraine died in debt (he does not seem to have had the financial acumen of his brother Franz Stefan), Joseph decided that he would not be bound by the detailed provisions of the will since all members of the imperial family had to obtain his consent as head of the family before making their wills, and Charles had not done so. Joseph decided to sell off Charles' property to pay his debts. I don't know what may have happened to his illegitimate family; they may have been provided for already, since as I understand it, Charles kept them very much under wraps in order not to offend Maria Theresa, and a public naming in a will would have done just that. Max Franz would have been provided for by Charles' death anyway because he would succeed to the Grand Mastership of the Teutonic Order (and the income) in Charles' place. Leopold was infuriated by Joseph's decision which he considered "unjust, despotic, absurd and outrageous'. I don't know if that was due to any bequest he or others in whom he was interested would have been deprived of, or whether - as usual - he was infuriated by Joseph's lack of consultation with his heir.
Why the annoyance of Mimi and Leopold and do you know who else in the family got money, estates and personal possessions from Maria Theresa?.....It was mentioned either here or at another thread that Louis XVIII later on asked about any inheritance due to Marie Antoinette from her mother and/or father but it appeared there was none?
I don't know about personal possessions - I assume all members of the family received mementos and there does not appear to be any contention about this.
Following Franz Stefan's death in 1765, Joseph and Maria Theresa set up a fund of 8 million florins to provide for Joseph's brothers and sisters. Although this fund was not in itself in Maria Theresa's will it's provisions are essential to explain her intentions in relation to her family after her death. Half of the fund was allocated in 1766 to Albert of Saxony as the husband of Marie Christine, in order for the couple to maintain a court in Pressburg. Marianne's appointment as Abbess in Prague, agreed before Franz Stefan's death I believe, was endowed with an annual income of 80,000 florins, a very handsome settlement. Each of the others were to receive 50,000 florins per annum until they were 'established'. In the case of the girls, by the time of Maria Theresa's death Maria Amalia was Duchess of Parma, Maria Caroline Queen of Naples, and Marie Antoinette Queen of France, so were due no further sums (it was not, strictly speaking, an inheritance and Louis XVIII's enquiry was ignorant, greedy and impertinent though of course a measure of his desperation at the time). Leopold was Grand Duke of Tuscany, Ferdinand was due to succeed as Duke of Modena, and Max Franz as Elector of Cologne. Ferdinand would continue to receive 50,000 florins until his father-in-law died. Although Max Franz would not become Archbishop of Cologne and Bishop of Münster until 1784, he was already Grand Master of the Teutonic Order by the time of Maria Theresa's death although her will might not have taken that into account since Charles died in July 1780 and Maria Theresa only 4 months later, hence possibly her bequest of the four estates mentioned earlier; although presumably, like Ferdinand, Max Franz would have been entitled to the 50,000 florins anyway until he entered into his establishment. Maria Theresa's will provided for Marianne to exchange her Prague convent for that of Klagenfurt as this was less expensive than Prague and more salubrious; her 50,000 florins would therefore go further. The will also provided for Maria Elisabeth to go to Innsbruck with 50,000 florins. Maria Theresa did not wish either Marianne or Maria Elisabeth to be forced to go to their convents but the money certainly helped Joseph to 'persuade' them to leave Vienna. The only parts of Maria Theresa's will relating to the imperial family which Joseph altered were the provision for Max Franz and as I indicated before, these did not seem to create any ill-will in themselves.
However, Maria Theresa had made very ample provision for the continuation of court pensions and other provisions for aristocratic imperial supporters which annoyed Joseph who was looking to recoup funds expended on Bavarian War and in general to reduce funds for the nobility. It seems it was chiefly in the ways he chose to take forward his court reforms which reduced or disregarded Maria Theresa's bequests which alienated Leopold, who was not consulted and who felt again Joseph was being high-handed and tactless.
Marie Christine was aggrieved at not receiving more under Maria Theresa's will - although she had already received half of the fund established for
all the imperial brothers and sisters - and Joseph took the opportunity to challenge some property arrangements Maria Theresa had made before her death in Marie Christine's favour, forcing Marie Christine to borrow some funds from Leopold to recompense Joseph. Leopold felt he had to lend Marie Christine the money, as she had made his son Charles her heir. However, it probably added to Leopold's irritation with Joseph.