When applying for a new job or fellowship, how to protect your ideas from being stolen by the lab you are applying to? If you want to change fields you have to explain the reason why you want to do that. In case of fellowship, you need some professor to host you and support your application, so you must contact him by email and explain to him what your idea is. 

What can you do, if your application failed, but the laboratory you wanted to work with, will carry on with your proposed idea/research? They have resources, people etc., while there is no such laboratory in your country so you are stuck. 

There is no legal reason why would they let you know how do they proceed, you do not know whether you should contact another laboratory, you cannot demand that you will be a co-author in case they are successfull.
I wouldn't worry about this, unless you have some reason to think you are particularly at risk for having an idea stolen.

Ideas are generally not the bottleneck in research.  Any experienced researcher will have many more ideas than there is time to investigate, and the hard part is actually carrying out the research well.  This means there is little or no incentive to steal untested ideas, especially in comparison with the risk of professional embarrassment from an accusation of theft.

When I've seen impressive ideas in graduate school or job applications, I've admired the applicant more than the idea.  Sometimes there have been cases where it would be exciting to work on this topic with the applicant, but
I've never seen a case where an initial idea was so obviously great in isolation that I wish I could take it up myself without involving the applicant.

That's not to say theft can't happen, but I don't think it's a likely or worrisome threat.  There's also not much you can do to prevent it in this scenario.  You can create enough documentation to prove that you had the idea yourself, but it's much harder to prove that the other person hadn't already had the same idea independently (which does happen on occasion).  You might be able to prove that someone behaved like a jerk by not telling you that they were competing with you, but you probably can't prove that they stole your idea.

Given that it's not a big risk and isn't easy to address, I'd recommend not worrying about it.