携帯を使いながら運転するって英語でなんて言うの？

At the wheel = driving a motor vehicle of some kind
Illegally.. = doing something which is against the law or contrary to current regulations.
"He's illegally selling drugs to school children."
Engaged on a call = making or receiving a telephone call.
'While in control of a motor vehicle.' This is a phrase taken from the wording of various traffic offences in the UK.
"The man was charged with being drunk whilst in control of a motor vehicle and sentenced to 3 months imprisonment."

Using your phone while driving is never recommended. In fact, it's illegal. Since you were distracted from driving while using a phone, an accident occurred.

Texting - Means to send someone a text message
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A: Have you heard about the huge car accident?
B: No. What happened?
A: John was texting and driving. As a result he collided with a truck.

"Being on a call while driving "
This means you are engaged n a phone call and driving at the same time.

Driving and texting
This means you are driving and writing/typing messages at the same time.

All of these expressions can be used to tell someone that you saw said person on their mobile device/phone while operating their vehicle.

To express that an accident happened as a result of someone being on their phone you may say that the accident was caused as a result of the driver texting while driving. Texting while driving is a well-known expression in English and is similar to that of drinking while driving . Both are in many countries considered a criminal offense.