You don't need to use the word "sandbox" but there may be a delay in indexing the redirected content based on how often your original site gets crawled. (The redirects that went into effect when seomoz changed to moz, for example seemed to happen right away, for example. others might take a month or more.

In your case, it sounds like you're moving from one domain to another?  If that's the case, then the redirect should be reflected in Google's search results as soon as the original domain is recrawled.

Being a word guy, I have a little bit different take on it:

'"Natural traffic" isn't a term that's widely used in SEO.  In fact, I'm not sure I've ever actually heard it used--ever.  "Organic traffic" derives from "organic search results", which were originally the algorithmic, non-paid search results before there was Local Search. Organic traffic is that traffic that comes to your site from non-paid, search results (and technically, also not from Local Search--although that might be debatable.)

Rexjoec,

It is considered duplicate content and the same holds true for www and non-www.  You can 301 forward the non-www to www and use rel=canonical to avoid the duplicate content issue with between http and https.