One can issue the SQL*Plus command SET TIMING ON to get wall-clock times, but one can’t take, for example, fetch time out of that trivially.
The AUTOTRACE setting, when used as SET AUTOTRACE TRACEONLY will suppress output, but still perform all of the work to satisfy the query and send the results back to SQL*Plus, which will suppress it.
Lastly, one can trace the SQL*Plus session, and manually calculate the time spent waiting on events which are client waits, such as “SQL*Net message to client”, “SQL*Net message from client”.