What is the ruling regarding the Hajj of the mother and the Hajj of the questioner after performing 'Umrah in Shawwal, then performing Hajj as individuals (mufridīn) from Riyadh?

Your Hajj is valid, and you are not required to offer a sacrifice (dam al-tamattu‘) according to many jurists, as long as you traveled after the ‘Umrah and departed from Makkah a distance that permits shortening prayers (masāfat qaṣr). Ibn Qudamah mentioned that one of the conditions for tamattu‘ that necessitates the hady (sacrifice) is that one should not travel a long distance between the ‘Umrah and Hajj, a distance at which prayers are shortened. This view was transmitted from ‘Ata’, al-Mughirah al-Madini, and Ishaq. Al-Shafi‘i holds that if one returns to the miqat, no sacrifice is due. The Hanafis believe that if one returns to their hometown, their mut‘ah (enjoyment of tamattu‘) is nullified. Malik holds that if one returns to their hometown or to a more distant place, their mut‘ah is nullified. In contrast, Al-Hasan and Ibn al-Mundhir believe that one remains a mutamatti‘ (one performing tamattu‘) even if they return to their country, based on the generality of Allah’s statement: {Then whoever performs 'Umrah [in the months of Hajj] before [performing] Hajj - what can be obtained easily of sacrifice}. However, ‘Umar and Ibn ‘Umar are of the opinion that if one departs and returns, they are not a mutamatti‘. Ibn ‘Uthaymeen holds that if one travels to their country after ‘Umrah during the months of Hajj and then returns in a state of ihram for Hajj, their tamattu‘ is interrupted.