How is the 儿 ending pronounced outside Beijing?
In MOST cases... Outside of Beijing, in texts, I believe the 儿 is still pronounced. But you can be sure that in spoken Chinese, it will never be pronounced (unless some kids are trying to mock the access by over accentuating it).
I said in MOST cases because there are some words that have simply been adopted by non-Beijing'ers and will always be pronounced with a 儿, and would sound weird otherwise.
Examples: Some Beijing phrases may be somewhat disseminated outside Beijing:
抠门儿 kōumér – stingy, miserly (may be used even outside Beijing) 

So whether you're from Beijing or not, you will always pronounce and write this word as 抠门儿 kōumér.
The closest I can come to describing this in American English is how we pronounce certain areas/things in the actual dialectical origin of that word. For example, we say &quot;New Orleans&quot; as someone from New Orleans would say it: &quot;Norlins&quot; even though we're not from New Orleans necessarily. If you say &quot;New Orleans&quot; and pronounce every syllable, it would sound weird to anyone... native or not to New Orleans.

EDIT: a better example is Louisville Kentucky. EVERYONE pronounces Louisville as &quot;Louville&quot;.