Is a girl considered a forbidden degree of kinship (mahram) to a young man, such that it is impermissible to propose to her, if his older paternal half-brother was nursed by the girl's maternal uncle, thus transferring the bond of brotherhood and making the girl as if she were his sister's daughter?

It is permissible for the young man to propose to the mentioned girl. The nursing of his brother with the girl's maternal uncle or her sisters does not affect this, as the prohibition relates to the nursling and those who nursed with him. As for the older brother, if he was nursed by the girl's grandmother, then he is her maternal uncle through suckling.