Regarding Google, as far as I understand that search engine workings (nobody really does, that's why there are SEO experts
), it will take short notice to Googlebots to see a lively website/forum at the new address, a couple of weeks maybe?, since they are linked one another on the main page. Both websites will coexist somehow in search, until the old forum will be acknowledged as no more active (that is, no more new posts). From that point onward, the new website will prevail on search results, unless you're searching for old arguments/posts.
Other search engines use more or less the same policy. The more active and updated is a website, the better is the positioning in searches.