I've only heard it around the house, first of all coming from the woods, which are fairly close by, about 70 yards or so, but since then I've heard it, I think, coming from the other side of the house towards the street, as if it's perched on a aerial or chimney but I can't see anything. The call is much slower and more monotone than the Nuthatch (one of my favourites they are) which did indeed sound more like a wee-wee-wee as suggested in your link. This is very much toot-toot-toot (if you whistle short and sharp yourself this is what it sounds like, almost non-birdlike, like I said, it genuinely sounded like kids messing about but it's clearly not that). I think the only time I've heard it is around this kind of time, spring/summer, and only for the last few years too.
I get usually the main kinds of garden birds on my feeders - mainly Sparrows, Starlings, Blue tits, Jackdaws, Robins, Dunnocks etc, the occasional Lesser spotted woodpecker, Greenfinch, Goldfinch and Sparrowhawks but nothing much more unusual than that, but do see and hear others when slightly further afield (but not this bird in question) but still can't place it. It's almost like it's a Blackbird, because of the tone/pitch, but with a broken beak or something and is only capable of making the same monotone whistle three or four times in a row. Proper got me this one has.