Very late response so I'm "necroing", but as I've seen a lot of posts saying that it doesn't mean anything, I have to disagree. My theory is that it has to do with the 4 elements, it is like that throughout silmarillion if you pay attention to it, and sometimes it's more explicit than implicit so I do think Tolkien did this very consciously in being specific with colours. Now this is understood in a way that isn't meant to be understood as if everything is purely separated as "x wizard is static", rather that all this is of Illuvatars mind and fully symbolical. I also made a summary of it, although not in english so it would be a hasssle to write it all, but I'll just mention the most important part; for instance the three silmarills in the end of silmarillion meet the 4 elements, one is lost to the fire and earth, one is lost in the sea (water) and one was sent up with Eärendil and his boat in the sky (air).

So back to the wizards, Radagast was the earth element, the blue wizards water, saruman air and the tricky part comes with gandalf, first grey and then white, but I later figured that he was in possession of one of the elven rings (the ones made by Sauron) which was the red one of the three, completing the circle. Him being grey could be because of him not willing to have any 'bonds' to anything, which was why he always was on the road traveling for instance. (I would have to read the Lord of the rings again to see if there is any more information about it there), but in Silmarillion and the chapter "of the rings of power and the third age" it is mentioned about the red ring, which is even called "The ring of fire". A quote from the end of that chapter, approximatley translated to enlglish (since it's translated in my language) when Círdan whom is a shipman (an allusion to Eärendil), who says in a quote approximately "my heart is with the sea", an allusion of the water element passing the fire to Gandalf, and: "Take this ring. Your patience and work will be heavy, but this will strengthen you in everything and protect you from tiredness, because this is the ring of fire, and with this you may reawaken hearts to its old boldness in a world that has become bleak". I read plenty of allusions here, for instance a premonition of the reawakening of Gandalf, the "bleak" as in Gandalfs colour "gray" at the same time as in the world is waging war against the darkness and losing hope/becoming dimmer/bleak etc.

And let's not forget that the fire element is often there as a symbol of the flame imperishable, which is the main source of existence from Illuvatar. And as the silmarills are described of the material they're made of, approximatley translated: "And this crystal was for the silmarills only what the body is for the children of Illuvatar; the vessel around the inner fire, that which is inside but yet in all its parts, and that which is life itself" (in the chapter 7 of Quenta Silmarillion).

Later Gandalf becomes the white as replacing Saruman (even though he is still Saruman the white but of ill nature, as you said: "all that is gold does not glitter").

That they are two blue wizards might be as a "compensation" for the missing "red"(fire) that has not been revealed yet, which as said would be in Gandalf's possession and only revealed in the end, which once again concludes a "process" when all elements comes in balance again (and this is after the one ring is destroyed), also as mentioned how Quenta Silmarillion ended, and several other places throughout which will be too long to sum up here as said but I hope it's somewhat understandable and that you got somewhat a new perspective on it. This is atleast my conclusion and it explains (to me atleast) everything better than any other theory which only focuses on the "material" aspects of the story so to speak, considering that Tolkien was a genius I strongly doubt that he wrote anything without a purpose, even if it's as "little" as the choice of colour.