Certainly a good question. I’m pretty sure I’m right in saying that the sound of the tune written in D coming out of your Bb whistle will in fact be the same tune only in Bb. Your fingerings, of course, remain “in D”. So the answer is “yes”.
This of course is the fiction of transposing instruments: you pretend that you’re playing “in D” by reading music in D and using the same fingering pattern no matter what whistle you’re playing. The only thing that changes is the relative pitch & key of the music you’re making.
You can continue to play the tune “in D”, and on your Bb Gen, by understanding that xxx xxx=Bb, and xxx xoo=D, and if you want to torture yourself by learning the Bb fingerings!
Cheers!