Is the money inherited by the mother, which the children have developed through trade, considered the property of the children, and do the sons of the deceased among them inherit their father's share, or does this money revert to the mother while she is still alive?

The money inherited by the mother is her property, and her permission to trade does not mean ownership of the money is transferred to her children.

The profits from the trade are considered according to the agreement. Either the children's work was a donation and the profit belongs to the mother (mudarabah (definition) with the owner providing the capital and the worker providing the labor, often with the worker being paid a fixed amount or a percentage of the profits), or all the profit belongs to the children and the money is a loan from them, or it is a partnership between the mother with her money and the children with their effort, or nothing was agreed upon.

The capital remains the property of the mother, and the children or the heirs of the deceased brother have no right to it.

If there is remaining profit in the case of a partnership or if all the profit belongs to the children, then the share of the deceased brother goes to his heirs.