As someone who has spent no more than a few short weeks of his sixty-plus years outside the boundaries of the continental US, it never occurred to me that the phrase "old-timey" would be considered silly or overly cute or whatever else one might mean by "twee." Only Brits would have that reaction.

Many aspects of culture differ on either side of the ocean. As noted earlier, "old time music" in the UK might mean music-hall/vaudeville, while in the US ~ even outside the "folk" subculture ~ the term is widely understood to define an entirely different genre of rural string music.

It seems, also, that in the UK, subsituting the word "timey" for "time" in this phrase is seen as unnecessarily cutesy or juvenile.

In the US, "old timey" is simply the native pronunciation of this term; the urbanized or more academically-correct phrase "old time" is more likely to be heard as an affectation than would be the good old traditional term "old timey."