New vehicle registration

When you purchase a car, is that car’s registration immediately linked to your name and address?

Basically, I think I drove in a bus lane by accident when it was in operation on the same week I got the car. However, the dealership had sent off the V5C form to officially register me I think and then I received it about 7-10 days later. So my question is if I did mess up at any point after I purchased would the council (or Tfl) know where to send me the letter (fine)?

I never received anything and it’s been more than two weeks. I just hope nothing has got lost.