In all likelihood, there wont be a problem. It is quite pointless for the first ministry to pursue the matter with the second ministry....especially when all are government services.

In all the situations which I have encountered, the most difficult part is getting the first party (e.g. HR) to initiate a request to hire you.....(can't find a better phrase to describe this process but u know what i mean) But once that is done, the rest will just quote this approval and everything should fall into place.

Usually, no one will contend against it. It is not as if you are a world class footballer where everyone is vying for your expertise. There are many others waiting in the queue.