I'm just back from a week of intensive recorder playing, taking classes and listening to concerts by professionals. I have the following thoughts:A note may not be in tune even though you have your fingers on the right holes. Fool around with the following:
1. Move your thumb closer and further from the thumb hole and note the effect.
2. On high notes, move the low fingers of the right hand closer and further from the open holes. See what happens. You will discover that an "unused" finger hovering over a hole can flatten a sharp note and make it sweet. I have just discovered, for example, that moving my third finger closer to an open hole takes the screech off my high E. Heaven!
3. For high notes which are too shrill, start adding fingers at the bottom of the whistle and see if they play in tune. You probably can't (and don't need to) do this for quick notes.
4. You can change the pitch of a note by the shape of your mouth and throat. Shape them as if you were singing the note, so that your whistle doesn't have to do all the work.
5. Can you whistle? If so, note the placement of your tongue when you do. Then, to get high notes, imitate that high, front position of your tongue.
6. Do you raise notes by pressing your thumbnail into the thumbhole? If so, try swivelling your nail so it goes across the top of the hole rather than the side. Sometimes that helps.
7. There are a lot of suggestions here, so print this out and work on them one at a time.
I also have two opinions to offer. If you want to make sweet music, forget the whistle and buy a more serious instrument, maybe a recorder. I don't believe that no one in Scotland or Ireland played the recorder in the olden days. The instrument is too good, too ubiquitous and too accessible for musicians not to have used it, especially those who were playing for hire, either at the Hall or for weddings, etc. I believe that today's emphasis on whistle is mostly marketing ("I'm just a destitute Celtic shepherd who is also a musical genius.")
I think the idea of spending 100 pounds on a whistle is unreasonable. To see why, walk around a hardware store and see if you can find any other thin-walled pipe with holes in it that costs anything like that.