A complete change of direction?

There is no slavish adherence to "direction". What you're doing is actually using the tiling method as intended, so feel free. That is the point of it, freedom from constraint by a single story line that well might peter out. Often, classic linear writing constrained by the outline (or in your case consciously dismissing that step entirely and jumping straight into the creative part) of a chosen story line ends in failure.

Instead, a change in direction, a change in order, characters, even trashing a complete plotline is allowed and even encouraged. It's brainstorming, first of all, and a different and more unconstrained way to accomplish the necessary functions of outlining without the constraints of the classic outline (my knowing you find outlining dreadful to do and the result an annoyance).

So, I'm not at all shocked over or worried about a complete about-face, a radical change, or the exploration of the notion of tiles created for a completely different story getting used in this one, such as Tinkletap (or the ginger tom story). Assembling tiles from something intended as totally different, written perhaps long ago, and mingled them with those you are creating now into an entirely new mosaic is well within the creative intent.