My question is: do I have to offer expiation or pay the diyah (blood money) or not?.
If the matter is as you say, and you were travelling under the speed limit, and you were unable to avoid hitting the bus, because it came out so suddenly and overturned in front of you, then there is no blame on you and you do not have to pay the diyah or offer any expiation, because you did not break the rules and you were not negligent.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked about a person whose car overturned, and his father who was travelling with him died – does he have to offer any expiation?
He said: It is essential to determine the cause of the accident; if it was caused by negligence or breaking the rules, then he has to pay the diyah and offer expiation. If it was not caused by breaking the rules or negligence on his part them he does not have to do anything.
Fataawa Islamiyyah, 3/357.
The Standing Committee was asked a similar question and replied:
If the driver was negligent or there is a reason that caused the accident such as his breaking the rules or driving too fast, or he was drowsy and so on, or he was negligent in checking his car or neglected some safety precaution, then he has to offer the expiation for killing, which is freeing a believing slave; if he cannot do that then he has to fast for two consecutive months as an act of repentance to Allaah. If he did not cause the accident in any way, then he does not have to do anything.
Fataawa Islamiyyah, 2/356
It says in a report of the Islamic Fiqh Council, issued during its eighth conference in 1414 AH (1993 CE) concerning situations in which a driver is exonerated of blame:
(a) If the accident happened as the result of reasons beyond his control, and he was unable to avoid it; this includes anything that is beyond man’s control.
(b) If it was mostly caused by the one who was harmed.
Majallat al-Majma’ al-Fiqhi, issue no 8, part 2, p. 372.
This applies in your case.
And Allaah knows best.