Question: On average how long does it take Apple to update/publish my new app/update on the App Store and iTunes website?

Answer: 
This link requires you to sign-in to your developer account:

<https://developer.apple.com/appstore/resources/approval/index.html#expect>

There is a box on the right with approval statistics - *most* update reviews are currently being completed within 8 business days.

E:

There are no longer statistics on that page (and it is no longer dev-account protected).

Statistics are now available without signing in: 

<https://developer.apple.com/support/appstore/#appReviewStatus>

There is a box on the left with % of new/updated apps reviewed in the past 5 business days.

There are also community efforts to continue tracking approval statistics by # of days:

<http://appreviewtimes.com/>

Keep in mind that community efforts rely on self-reporting of approval times.
usually around 8-10 days from what I've experiences (and according to this site)

<http://reviewtimes.shinydevelopment.com>