What is the ruling on someone who missed the Dhuhr prayer until the 'Asr call to prayer while they were in a car and unable to stop, so they performed tayammum and prayed Dhuhr in the car, then prayed 'Asr 15 minutes after arriving home?

It is not permissible to delay the Dhuhr prayer without a valid excuse until the time for the Asr prayer begins. Whoever does so without a legitimate Shariah excuse has committed a grave sin and must repent. Scholars differ on whether performing the Dhuhr prayer after the Asr time has begun is considered timely performance (adaa’) or making up for a missed prayer (qadaa’). The majority of scholars hold that it is qadaa’, while the Malikis consider it adaa’. However, they all agree that delaying it without an excuse is a sin.

As for praying in a car, the general rule is that it is not permissible, unless there is an excuse such as fear for one's life or property, or being separated from one's companions. In such cases, it is permissible to pray on a riding animal (or in a car, by analogy) to honor the time of prayer, but according to the majority of scholars, the prayer must be repeated. Therefore, if the questioner prayed in the car due to fear, what she did was correct, but she should repeat the Dhuhr prayer as a precaution and to avoid scholarly dispute.