You would issue proceedings against the driver and the employer. Unless you have experience of bringing personal injury claims, you would actually be better using one of the firms of solicitors that deals with personal injury claims on a no win no fee basis. There is no advantage to doing it yourself because the costs of the claim will be recovered from the fault party.
Trauma and stress claims are not the easiest claims to bring. From what you have said, you do have a claim but it is not going to be easy.
For it to be actionable, you need to have a recognised psychiatric condition such as PTSD. You will need a report from a psychiatrist to support that and that report alone will normally cost between £800 and £1000 plus VAT. Without that report, your claim would fail.
The solicitor will write to the bus company inviting offers to settle and ultimately, if an offer is not forthcoming they will issue proceedings for you. They will also arrange insurance so that if this does end up in court, and you lose friendly reason, you would not have to pay the other side’s legal costs out of your own pocket.
For that latter reason alone, it is something that I would not recommend that you try to deal with yourself.
Can I clarify anything for you?