I’m a beginner at the whistle but I’ll throw in my two cents.
Moving the position of the whistle in your mouth can certainly affect the sound—try it out and see.
I think tapping your foot can be done without moving your instrument. I am able to do that. You can just tap your toes too. In classical music, people just don’t tap their feet the way they do in traditional music, in my observations. I don’t know of any special guarantee that a person taps his/her foot consistently though, if the brain does not have the tune right in the first place. I do enjoy tapping my foot, but my foot is going with my brain, my brain is not going with my foot.
You will find that once you have learned the notes of the tune very well, both in terms of hitting them and in terms of the correct rhythm, you will be able to keep the correct time without the metronome or tapping.
You are using the metronome to train yourself to play the tune correctly. You won’t need to use it once you have really learned the tune. After playing with the metronome, you will be able to hear if you are going off—wrong rhythm, speeding up, etc. That would mean you need to work with the metronome again at a slower speed.
I use the metronome at the beginning of learning a tune, especially if I notice I am having some trouble. I get the tune so that I can play through it properly, even if quite slowly. Then I practice without the metronome. If I hear that things are not quite right, I go back to it. But it is more fun to play without the metronome—you don’t want that ticking to always be there. So, even when you play alone, you want to get trained and then play without it. Does that make sense? The metronome is not going to be your constant companion on a tune.
If you are playing with a group, I guess you do need to agree on who is setting the tempo.
Maybe that is the drummer, maybe that is someone else. Someone else will have to give the band question an answer.
I know many people think metronomes are not a good tool. Some people have grown up listening to the type of music they are playing, and so perhaps they have been trained by listening. Some have people around that can tell them when they sound weird. I think that whatever helps a person get on a bit is good.