For the training I've done on the ride, the go around is normally given off an RNAV or other non-precision circling approach, which is hand flown from the MDA through to the go around (normally followed by some form of engine malfunction). The ILS is then done as a single-engine approach which, for most of the types I've flown, the autopilot is not approved to be used for - so you have to hand fly it. But that's about it - maybe, 10 minutes of an hour and a half ride are hand flown - assuming you don't have to demonstrate stalls or steep turns.
It would be nice to see a little more hand flying though, which is why we've started incorporating autopilot failures into our training. At least that way we know they're getting the exposure.