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

Answer: 
It depends on a human at Apple reviewing your app. If you'd made a few small graphics changes then it won't take very long, if you've added stuff like access to the address book or whatever they may look more closely. And it may take a while for them to get to yours if lots of other apps have been submitted recently too. So you might as well ask "How long is a piece of string?"
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.