Does anybody know how to set the permission or simply achieve a way of displaying the weather module in a block for unregistered users like on this page? As is seems by default on my installtion users need to be logged in.

Hello,
the weather module and the weather block fully obey PostNuke permissions.
For the block to display a visitor (registered or unregistered) need read access
For the module to display the visitor need read access but they won't be able to change their prefered city, for that they need overview access.

So if you haven't set any group rule in Administration->Permissions that limit read access of unregistered user, the block should be visible by anyone. If this is not the case, can you post ALL your permissions on this forum and I should be able to diagnose the problem.