The biggest problem with desalination of water worldwide likely would be microplastics and other pollution

Where does all liquid waste of every kind on earth wind up?

That's right it winds up in the ocean eventually if it is liquid based usually.

So, no matter how toxic something is or even if it is radiated by nuclear waste it all winds up in the ocean. 

Then you have scuttled nuclear submarines and ships that are scuttled out to sea various places which are also spreading radiation through the oceans. Then you have Fukushima meltdowns into the earh (3 of them) which will spew radiation out to sea there for 25,000 to 50,000 years non-stop from one of the meltdowns that was reprocessing weapons grade plutonium.

Can desalination remove all pollution from the water?

I'm not sure, especially when just breathing now you are breathing in microparticles of plastic waste around the world too. 

I was writing recently that the genie of plastic and the genie of Radiation cannot be put back into the bottle. They are both here for all time now on into the far far future of earth. And everything that lives on earth will have to somehow cope with that.