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I am on Tier 1.

I am expecting a baby and it turns out we may need to travel internationally for work 5-6 weeks after his expected date of birth.

In this case, I will need to apply for T1 dependent visa for him.
The difficulty is that I have so little time to take care of it all.
First, what documents do I need?
Particularly, do I need baby's passport? It takes some time to get his passport.
Do I need to send along my passport? If I do, we cannot travel obviously if the procedure takes time.
Do you have any idea how long it takes by post? The official 8-week limit sounds pretty long.
Is it possible to have a priority application for dependent-only applications? If so, how much?

There is so much information online and in this forum, and I understand that the regulations and fees change all the time. So I am pretty confused. Any help is much appreciated.

First, what documents do I need?

Read the Tier 1 dependent guidance notes.

Particularly, do I need baby's passport? It takes some time to get his passport.

Yes you need a passport.

Do I need to send along my passport? If I do, we cannot travel obviously if the procedure takes time.

Yes however a copy should be fine.

Do you have any idea how long it takes by post? The official 8-week limit sounds pretty long.

Up to 8 weeks, sometimes longer.

Is it possible to have a priority application for dependent-only applications? If so, how much?

You can apply using premium service for a same day decision. Costs an extra £610 on top of the visa fee and IHS fees.

You could also apply for the childs visa from home country if you wish to.