You definitely do not need a car in Birmingham. The transport links are very good. The University of Birmingham itself has a train station on campus and many bus stops that run frequently. You will also benefit from a student discount with a valid student ID card.
Closer to the city centre, where you will find BCU, we have lots of buses and trains but we also now have a tram that takes you from Edgbaston through the city centre to Wolverhampton!
The only time are car might be useful is if your course involves placement. You can be sent quite far sometimes so it is a good idea to check whether this is something that might affect you.
Also, insurance is crazy expensive so you need to weigh up the cost of having your car in Birmingham. Most student accommodations charge you to park your car on-site too.