% pandoc -f mediawiki -t native
''italic''
[ Para [ Emph [ Str "italic" ] ] ]
While mismatched input format—syntax does not work, which is also expected:
% pandoc -f twiki -t native
''italic'"
[ Para [ Str "''italic'\"" ] ]
% pandoc -f mediawiki -t native
_italic_
[ Para [ Str "_italic_" ] ]
It seems your wiki files are mediawiki instead of twiki, in which case you should specify mediawiki as input format. In case you have a mixture of both the presence of double and triple single quotes is a pretty sure indicator that you have a mediawiki file.