Can a substituted cyclohexane progress through SN1?
I think it's difficult to say for sure without actually making experiments. It should depend a lot on the conformation of the reactant, the substituents, the leaving group and also on the solvent. Nevertheless, I would imagine that it should be possible in some cases if you have a good leaving group. 

In the case of a substituted cis-decalin, for example (I suppose you actually meant to refer to cis because in the trans isomer both hydrogens are stuck in axial positions), the elimination of the leaving group must indeed induce some strain, but maybe not that much. The structure could adapt with small changes in angles all over the molecule and both cycles would keep chair-like conformations. A tertiary carbocation would also be quite favorable.

So yes, in my opinion it should be possible in some cases, specially if heating is employed or you have a good leaving group. However, I should reinforce the fact that it really depends on many factors. This type of system can get quite complex if you consider the influence of the solvent, the influence of the substituents and the possibility of side reactions, including eliminations reactions, which commonly compete with substitution (on the presence of bases).