This is certainly not an area of my specialty, so most of what follows is the simple understanding that I’ve gleaned from a pretty complex subject, and the question itself is difficult to interpret when you ask for “subatomic” and “not quantum” in the same question. Essentially, isn’t subatomic the essence of quantum physics? That being so – if it is – how can you exclude quantum physics from a discussion of subatomic transistors / switching?
So… we’re (I’m using the royal we, here, because I’m certainly not involved) working on quantum computing now, where qubits can do three-position switching instead of the current binary switching that we’re (ahem) used to with current computing. That, all by itself, will be a major leap in information processing and storage. If you’re talking about “transistors”, obviously you’re more interested in processing, but since processing is also heavily dependent upon the information itself, then storage also matters.
Continuing on the quantum line, since quantum entanglement across distance seems to be a real thing, then if we can understand and control that and “read” results, it may paradoxically become possible to make the subatomic switches (transistors, in effect) that you’re thinking about, but spaced far wider than current microchips, which could resolve a lot of the power and heat issues with crowding more chips in less space. So, think “distributed processing at a distance, but with zero time lag”.
And as @LuckyGuy suggests, if we can someday knowledgeably, routinely and with some semblance of control operate along more axes than 3D + time, or if we can add dimensions to time itself in ways that I can barely imagine and probably can’t describe (how about this: serial processing that happens at right angles to the timeline of past-present-future that we live along, so that a long process could be completed along a separate time axis in such a way that it appears instantaneous to our experience) – then who knows what is possible?