The uke is a Hawaiian instrument, and ---as I understand it---was introduced to Hawaii by Portugese settlers in the last quarter of the 19th century.  
 
I think we can safely say that both the uke and the cuk came from the same Portugese instrument, only at widely separated points in time. 
 
Yes, I agree that the cavaquinho could well have been the original instrument used in fado. 
 
As for similar sounding words, yes, this can cause a lot of amusement. Even the cuk that they use in kroncong is the same sound as the Australian colloquial for a chicken:- chook.
		 
		
		
		
		
		
		
		
	 |