How about a concrete example? Take M to be the closed unit disc in the complex plane. Then its boundary is the unit circle. Taking U = {re^it: r>1/2}, we get a diffeomorphism of U with S^1 x [0,1) via re^it > (e^it, 2(1r)).
The problem with that is it is too specific to the example and doesn't really shed light on the general case. So... Here's another construction that I think captures the difficulty of the general case better:
Let E = {e^it + (1)s: pi/3 < t < pi/3, 0 <= s < 1}, N = iE, W = E, S = iE. Clearly, E is diffeomorphic to {e^it: pi/3 < t < pi/3} x [0, 1) and similarly for the other three; and the four together cover S^1. But is it possible to stitch the four diffeomorphisms together to get a single diffeomorphism of E U N U W U S with S^1 x [0,1)?
