I have always considered that the term "old timey" was probably used by a few older musicians from the south when describing the music from way back.
The problem in the UK with telling people you play old time music is that they tend to think of Old Time Music Hall (Vaudeville to you folks across the pond).
Personally I have no problem with the term "old timey" any more than when a southern musician will tell me that he "kindly plays it like this". It's purely a use of language and that changes all the time. I can no longer tell people that I like folk music, jazz or old time music without having to clarify what it is and not what they think it is.
As for clarifying "twee" I think that the Plastic Deer and birds that decorate the lawns in front of houses in the Appalachians for instance would be prime candidates for that description.

Hoot