The dowry (mahr) is an obligatory part of the marriage contract, as stipulated in the verse (interpretation of the meaning):
“All others are lawful, provided you seek (them in marriage) with Mahr (bridal-money given by the husband to his wife at the time of marriage) from your property, desiring chastity, not committing illegal sexual intercourse, so with those of whom you have enjoyed sexual relations, give them their Mahr as prescribed”
It is Sunnah to stipulate the dowry in the marriage contract, but if it is not stipulated, the marriage is still valid and the wife should be given a dowry like that of other women like her, who share the same characteristics according to which the rates of dowries may differ.
The dowry should be known; if they agree to an unspecified dowry, that is not valid, and the wife should be given a dowry like that of her peers.
Ibn Qudaamah (may Allaah have mercy on him) said: No dowry is valid unless it is specified. This is the view of al-Shaafa’i. End quote from al-Mughni, 7/168
It says in Ma’oonat Ooli al-Nuha (9/192): It is essential that the dowry be known. Then he said: That applies to everything of which the value is unknown and the way of obtaining it is unknown; it is not valid as a dowry, and there is no difference of scholarly opinion on this point. End quote.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
If he gives her something unknown as a dowry, that is not permissible, and she should have a dowry like that of her peers. End quote from al-Sharh al-Mumti’, 5/302
What is mentioned in the question about the uncle refusing to stipulate a dowry and asking for the wedding expenses of his nephew to be combined with those of his son, then shared equally, makes the dowry unknown, because it is not known how much his son’s wedding expenses will be.
Based on this, this wife should be given a dowry like that of other women who are like her, if the marriage contract has been done as you describe. If the marriage contract has not yet been done, then a dowry should be specified for her, and it is not permissible to do what your uncle is asking, because that will lead to the dowry being unknown.
And Allaah knows best.