it doesn't matter who files for copyright first if one group is without a doubt the origin of whatever is attempted for copyright, because all your original work is technically copyrighted the instant you make it, filing for the copyright is just to prove when you made it
(edit: errr, trademarks... i think they are the same though)
A trademark is used to protect a word, symbol, device, or name that is used for the purpose of trading goods. The trademark indicates the source of goods and distinguishes them from the goods of others. A trademark may also be used to prevent others from using a mark that might be confused with another; trademarks, however, do not prevent other people or businesses from producing the same product or services under a different mark.
Quoted form http://www.wisegeek.com/what-is-the-difference-between-a-copyright-trademark-and-patent.htm
It all comes down to whether the patent office feels that granting trademarks to both U.S. Navy and Disney would cause consumer confusion, Wilentz said, in which case the patent office would give priority to the entity that filed its application first -- Disney.
And that is from the article.
I think that Disney would get the rights to the trademark, because they filed for it first.