I'm of the opinion that tweeness tend to largely lie in the material not the voice. There a very few really good versions of some of the 'twee' victorian scottish songs. Shortbread box stuff. However, there are a lot of songs which loose their power and could be interpreted as twee depending on the delivery of the singer. As a female singer with a reasonbaly high voice at the top of my range and possibly a tendance to 'wisp' ( especially if I'm not got the breathing right), I am always aware I could sound 'twee'. Or worse - get the breathing right - give it full belt and sound operatic !
Another major contibution to tweeness which has not been touched on is the accomaniment. How many powerful songs are plunged it tweeness not by the singer but by overwrought accomaniment ?