Clockwork parts? No. Parts that select from a combination of inputs? Yes.

Computability theory demands repeated combination/selection and memory. It doesn’t otherwise care how this is accomplished.

In principle? No question.

In practice? No chance.

How would a clockwork model of the mind explain the phenomenon of consciousness (as opposed to computation)?

The idea is that consciousness is what a planning algorithm feels like from the inside. A planning algorithm requires representations of the entities relevant to planning, e.g. various objects in the world, relevant laws, internal states such as emotions, motivations, etc. It is these representations from an internal perspective that have something it is like to have them. But the fact that this planning algorithm is constructed out of mechanical components is irrelevant.

Computation is just an abstract description of the behavior of some classes of systems. But it doesn't exhaust the set of possible descriptions for a system. Some subset of computational systems might also have a description in terms of consciousness.

Your apparent consciousness is the illusion of control over your subconscious mind.