First of all - git submodule isn't working for me, so I cloned the dependencies by myself.
Instead of working around a problem, you should really try fixing it.
The submodule declarations contain more than just a repository name/source, they also contain branches/versions. Indeed, SFNUL doesn't use the master version of ASIO, so whatever you cloned probably doesn't work out of the box like the submodule does.
Secondly - had some problems with mingw headers, but found patch which fixed it.
This sounds even more troubling... What kind of a patch was this? And what did/didn't work? If anything doesn't work, at least ask for assistance before applying random patches to the base installation. They might end up breaking more things than they fix.
I'm using TDM GCC 4.9.2 DW2 and I successfully built SFML, SFGUI and all other goodness under it, so what's wrong with SFNUL?
There is nothing wrong with SFNUL, but there are many things wrong with your development environment as stated above.
Also, doesn't that whole "asio" need to be built? It's not included in CMakeLists.
No. Using the submodule that I picked out, ASIO can be used header-only. I have no idea what repository/branch you are on right now, but I can't guarantee that something you cloned yourself will work with SFNUL.
Rewind (reinstall MinGW if necessary), and go back to the "git submodule doesn't work for me" stage and provide some error information. That is a starting point that we will be able to work with.