Determine Oracle session client character set?
I am a little doubtful that this is exactly what you are looking for, but
host echo %nls_lang%;

ENGLISH_UNITED KINGDOM.WE8ISO8859P1

shows the client nls_lang environment variable on the client.
I don't think there will be a SQL query you can run to give the 'current' setting because AFAIK the server is not aware of what translation is done client-side, so any command to show the current setting will have to be native to the client - I used SQL Developer for the above command, but I assume it will work the same in SQL*Plus
--edit
from AskTom:

only the client knows their character set as well -- it is not available &quot;in the
database&quot;

and

the character set describes what is stored in database.
the client makes their desired translated to character know [sic] to the database via the NLS_LANG
settting.

If you were on 11.1+, you might have some joy with v$session_connect_info, because:

This information is pushed by OCI to the server ats login time.

But I discovered it would still depend on how you are connecting, eg from the JDBC Thin Driver you aren't using OCI and so the information isn't pushed