Praise be to Allah.
It is not stipulated in the case of tamattu‘ that the two rituals (‘Umrah and Hajj) be done on behalf of one individual. If a person does ‘Umrah on his own behalf and Hajj on behalf of someone else, or vice versa, or he does that on behalf of two different people, all of that is permissible and he has to offer the hadiy (sacrifice) of tamattu‘.
It says in al-Mawsoo‘ah al-Fiqhiyyah (14/11): It makes no difference (with regard to offering the sacrifice) whether the two rituals are done on behalf of one person, or he does ‘Umrah on his own behalf and Hajj on behalf of someone else, or vice versa, or he does that on behalf of two different people; he still has to offer the sacrifice of tamattu‘, because of the apparent meaning of the verse. This is according to the majority of fuqaha’. However, the Maalikis said: There is some difference of opinion in the case where both Hajj and ‘Umrah are done on behalf of one person, where the issue is not clear-cut. Ibn al-Haajib said: The most well-known view is that he does have to offer the sacrifice.
End quote. See also at-Taaj wa’l-Ikleel, 4/83; Sharh al-Muhadhdhab, 7/176; Kashshaaf al-Qinaa‘, 2/414
In the fatwas of the scholars of the Standing Committee (11/58) it says: It is permissible to perform Hajj on behalf of one who is deceased or on behalf of one who is still alive but is not able to do Hajj. But it is not permissible for a person to do Hajj once and make it on behalf of two people, because Hajj can only be done on behalf of one person at a time, and the same applies to ‘Umrah.
But if he does Hajj on behalf of one person and ‘Umrah on behalf of another in the same year, that is acceptable if the pilgrim has already done Hajj and ‘Umrah on his own behalf. End quote.
And Allah knows best.