Oracle 12.1.0.2.0, 64-bit client.
I believe I found the cause of the problem. My system's ORACLE_HOME was set to the path of an earlier (also installed) version of the client:
C:\oracle\product\11.1.0.6
I changed this to the path of the 12c client:
C:\oracle\product\12.1.0.2
And now Toad connects as expected.
Thanks very much! Hopefully your tips and this information will help other Toad users.