Probably close this thread.
By replacing 10^x_ with E^(Hold[Log[10] x) and surrounding with ReleaseHold, it evaluates.
ReleaseHold [
TransformedDistribution[ x = 10^(u /10 ), u [Distributed] NormalDistribution[ mu, sigma] ] ] //. {10^x_ :> E^( Hold[Log[10]] x) }
]
Of course, I'd still be interested in a good explanation as to why it works this way so as perhaps avoid the trial and error approach I used this time.