When you first make a connection to the new LocalDB edition of SQL Server Express, the system files, etc. that are required for a new version are spun up. (The system files such as the master database files, etc. end up in C:\Users\<username>\AppData\Local\Microsoft\Microsoft SQL Server Local DB\Instances\LocalDBApp1) That can take a while on a slower machine, so this means that the default connection timeout of 30 seconds (in most client libraries) could be exceeded.
To avoid this hit on the first connection, you can create the required instance of LocalDB beforehand using the SqlLocalDB.exe utility, like this:
<path to binaries>\SqlLocalDB.exe create InstanceName
You can also specify the required version of SQL Server and ask to start the instance like this:
<path to binaries>\SqlLocalDB.exe create InstanceName 11.0 -s
Your application should then connect quickly, even on the first connection.
SqlLocalDB documentation is starting to appear now. Documentation on the utility is here: http://msdn.microsoft.com/en-us/library/hh212961(v=sql.110).aspx.