Is this dialogue OK?
(by car)
Person A - Why are you driving so fast? / Why are you going so fast?
Person B - Lest we are late.And these two?
Why have you speeded up? I'm scared.
Why have you accelerated? I'm scared.

Question

Is this dialogue OK?
(by car)
Person A  - Why are you driving so fast? / Why are you going so fast?
Person B  - Lest we are late.And these two?
Why have you speeded up?  I'm scared.
Why have you accelerated?  I'm scared.

Answer

Lest is not used as often as it used to be, and it is not an answer to a question. It is saying I (you) am/are going fast in case we may be late. 
"I have to hurry (sped up) I/we may be late" = a more modern sentence.

Answer

Most people don't use the word lest anymore. You would more likely say "we are running a little late" or "because we are a little late".
Changed speeded to sped