Laurence, I know exactly what you mean.
Here is what I think:
1. It is not a matter of dialect. Even if we take the pronunciations of a broadcaster or an actor with a standard accent, the way that he does w in 我 is quite different from the way he does w in 威。
2. You have correctly spotted a real difference.
3. In fact, the Chinese w, whether in 我 or in 威, requires the lips to be positioned differently from the white Anglo Saxon w.
4. The w in 我 starts from much more of a "kissing fish" lip position, in order that the finishing o may be pronounced fully (说得满) and correctly. At the starting position of 我, the lips are pushed much more forward than the English w. This w is much more exaggerated than the English w.
5. The w in 威 starts from a more retracted lip position that is akin to the English v lip position EXCEPT that the upper teeth do not extend over the lower lip. The amplitude of the lip movement from start to finish us much smaller than in 我. This w is much more restrained than the English w. The "ei" sound of 威 is also pronounced from a much more compressed mouth cavity than the English "way". The English "way" is pronounced much closer to the throat. The Chinese "wei" finishes right in the middle of the mouth cavity. This finishing position of the Chinese "wei" means that the starting consonant of w is less overtly forward in order for the finish to be sufficiently compressed, with the finishing "i" ("ee" in English) quite fully pronounced.
6. We have to bear in mind that the English alphabet is a "transplant" that has been grafted onto Chinese sounds. In an ideal world, there should be two different symbols for the two starting consonants of 我 and 威, and neither of these should be the English w.
7. You are already ahead when your ears and eyes are good enough to have spotted the physical sound difference, though on paper it is the same w.