I'm not sure never really thought about it, i just thought it was that style of clothing peg that was called "dolly peg" as you'd see people draw faces on them.
Check theses out
Done a bit of research after noting replies -
Seagull/Hammockman - I think that this is the tub and the 'posher' or dolly that you mention.
View attachment 45299
It appears that the term 'Dolly' was applied to any of the items - tub/posher/pegs etc
More info around this site http://www.oldandinteresting.com/washing-dollies.aspx
Just thinking ahead for the wife's Christmas presents.