早くして！間に合わないかもしれないよ。バス行っちゃうよ！って英語でなんて言うの？

"Leave without" is a common way to say things like this. For example:

A: Meet us at the Family Mart at 7:00.
B: Sorry, I'm going to be about 10 minutes late.
A: If you aren't here by 7:15, we will leave without you!

The bus will leave without us. = 私たちを待たずにバスは出発する。

Miss the bus = not be able to get on the bus because it started its journey already.
"Let's ......" This is a way of suggesting that you do something with someone else.
"Let's go to the cinema."
"Lets have dinner together this evening."
"Let's not argue!"

"Let's hurry up so we won't miss the bus!"
This is what you can say when you are helping your child to finish whatever he/she is doing so that they won't miss the bus.

" Finish up quickly! I don't want you to miss the bus!."
You can say this when the child is old enough to prepare themselves without your help and they need to finish whatever they doing to catch the bus.

Your child might be struggling with tying her shoe laces or could be looking for his/her misplaced pencil case and you are afraid this might cause him/her to be late for the bus.
So, you may say:
Hurry up so you can catch the bus.
or
Please hurry otherwise you will miss the bus.
or
Be very quick otherwise the bus will leave you behind.

You can tell your child "Hurry up or we'll miss the bus". Hurry up means do something quickly.
A colloquial way of saying hurry up in English is to say "get a wriggle on". Wriggle is a word to describe quick, jerky movements.
For example" If you don't get a wriggle on, you will miss the bus!"