The sentence doesn’t make sense. Similar examples with meaning: I don’t know whether they will move to London. (I’m ignorant) I don’t know that they will move to London. (There’s doubt in my mind that they will move) I didn’t know that they will move to London. (They will move but I wasn’t aware of it. Or ‘I was ignorant of the fact that they will move.’)

In this sentence, THAT seems to be a conjunction. It joins the two clauses. I'm ignorant + they will move to London. However, as Dan says, it is hard to say as the sentence isn't correct and the intended meaning is not quite clear. Do you want to say,"I did not know (that) they plan to move to London"? THAT is a conjunction in the above sentence. It is optional. You can use it if you want to, but it's not essential

In my opinion this sentence is not correct. It should be something like: I'm ignorant of the fact that they will move to London. Or maybe the sentence is trying to say something else entirely? Even corrected, the original sentence doesn't seem to make sense.