Ink I am no expert. But all the experts say that Uilliam/Uilleam is the Gaelic (Scottish or Irish) version of William. I'm prepared to believe them.
I absolutely agree that some Gaelic names have no translation into English, but that does not mean that there have not been borrowings between languages over the centures.
For example, Charles (Tearlach in Gaelic) originated from a Germanic word, Carl/Karl translated into Latin (Carolus) and was then spread throughout Europe by the institutional church and by royal conquerors. In Scotland and Ireland it was pronouced Tearlach. (In western Scotland, that sounds rather like Charlie: Tchar-lochh.)
Similarly Seamas (James) has Hebrew origins. It is the English version, via French, of the Old Testament name Jacob. In Gaelic it becomes (through pronounciation) Seamas.
The Wikipedia article I linked to explains in detail the spread of 'William'.
Over the years, names like Seamus and Tearlach and Uilliam become identified with Gaelic culture. Today, they have indeed become Irish, or Scottish. I really, really wasn't wishing to say otherwise. I was just wishing to point to their ultimate origins.