Cannot access MySQL database after moving files in data directory
Ok, the problem was because I didn't copy the ibdata* file. You may expect this file to be located within the respective database folders just like the other related files, but you would be wrong.

My ibdata file was located at mysql/data/ibdata1; as soon as I copied that to the new location everything was good again.

Note however I have read of this file being located in different locations, so your mileage may vary.