You've got a map from $[0, 1] \times \mathbb R$ onto the Mobius bundle, so I'm going to give it a name -- $f$ -- and identify each point in the Mobius bundle as $f(x, t)$ for some $x$ and $t$. The only place that's ambiguous is that certain points have two names: $f(0, t)$ is the same point as $f(1, -t)$. But I'll make a rule: I'll always use the name that has a $0$ as its first argument. (IF you like, I've restricted the domain of $f$ to $[0, 1) \times \mathbb R$.) That means that $F$ provides a 1-1 correspondence between its domain and the Mobius bundle. (The correspondence is NOT continuous at the two "ends" of $[0, 1]$, but at least it gives me a unique way to refer to every point of the total space of the bundle.)
I'm going to define a map $H$ from a subset of the domain of $f$ to $ V \times \mathbb R$ (in the language of your assignment).
$$
H : (0, 1) \times \mathbb R \to V : (x, t) \mapsto (e^{2\pi i x}, t)
$$
I'm going to define another map $K$ from a subset of the domain of $f$ to $U \times \mathbb R$:
$$
K : [0, \frac{1}{2}) \cup (\frac{1}{2}, 1) \to U \times \mathbb R: (x, t) \mapsto
\begin{cases}
(e^{2\pi i x}, t) & \text{for $x < \frac{1}{2}$} \\
(e^{2\pi i x}, -t) & \text{for $x > \frac{1}{2}$}
\end{cases}
$$
These two maps together map the total space of the first bundle to the total space of the second in a way that respects the gluing maps on each side.
I believe that this is what you need.
But you may be saying "Wait, that's a map, but how can I know it's a smooth isomorphism of bundles?" OK. Here goes.
First, I'm going to place two coordinate charts on the base space of the Mobius bundle. Let me call that bundle $M$, OK? Once again, I'll be using the map $f$ to identify points in $M$.
The first chart's domain is
$$
A = (0, 1) .
$$
The coordinate map $\phi_A: A \to \mathbb R$ is exactly $x \mapsto x$.
The second chart's domain is
$$
B = [0, \frac{1}{2}) \cup (\frac{1}{2}, 1)] / \sim
$$
And the coordinate map $\phi_B$ is slightly more complicated:
$$
\phi_B( [x] ) = \begin{cases}
x & \text{for $0 < x < \frac{1}{2}$} \\
0 & \text{for $[x] = [0] = [1]$} \\
x-1 & \text{for $\frac{1}{2} < x < 1$}
\end{cases}
$$
The image of $\phi_B$ is the interval $(-\frac{1}{2}, \frac{1}{2})$.
And now I'm going to build coordinate charts on the bundle; I'll call these $\psi_A$ and $\psi_B$, because one of them will trivialize the portion of the bundle over $A$, and the other will trivialize the portion of the bundle over $B$. Once again, I'll use the labelling of points of the bundle provided by $f$. Here goes:
$$
\psi_A : \pi^{-1}(A) \to (0, 1) \times \mathbb R : f(x, t) \mapsto (x, t).
$$
Clear? $\psi_A$ is just the inverse of the map $f$, with its domain somewhat restricted.
Now for $\psi_B$.
$$
\psi_B : \pi^{-1}(B) \to (-\frac{1}{2}, \frac{1}{2}) \times \mathbb R : f(x, t) \mapsto
\begin{cases}
(x, t) & \text{for $x < \frac{1}{2}$} \\
(x, -t) & \text{for $x > \frac{1}{2}$}
\end{cases}.
$$
Now let $AB$ denote the intersection of $A$ and $B$.
The transition function $\tau_{AB} : \psi_A(AB) \to \psi_B(AB)$ is once again defined by cases. I'll write it down, and then you have to check that it is, in fact, just $\psi_B \circ \psi_A^{-1}$ on the domain $\psi_A(AB)$.
What's $\psi_A(AB)$? It's $((0, \frac{1}{2}) \cup (\frac{1}{2}, 1)) \times \mathbb R$.
What's $\psi_B(AB)$? It's $((-frac{1}{2}, 0) \cup (0, \frac{1}{2})) \times \mathbb R$.
Now let's write down the map between them.
$$
\tau_{AB}(x, t) =
\begin{cases}
(x, t) & 0 < x < \frac{1}{2} \\
(x, -t) & \frac{1}{2} < x < 1
\end{cases}
$$
Whew!
Now you've got a similar structure on the other bundle: charts $U$ and $V$, coordinate functions $\phi_U$ and $\phi_V$. (I'd recommend $\phi_V(e^{2\pi ix}) = x$ for $ -\frac{1}{2} < x < \frac{1}{2}$, and something similar for $\phi_U$.) You can also write down the local trivializations $\psi_U$ and $\psi_V$, which are really pretty simple, and the transition function, which will look a lot like the $\tau_{AB}$ that I wrote down.
Then all that's left to do is to show that the function defined by $H$ and $K$ plays nice with these coordinate charts and transition functions (i.e., check the things that Erno suggested in his answer to your previous question).
If you write out $\phi_U, \phi_V,$ and $\tau_{UV}$ here, I'll go ahead and respond with something about checking those conditions on $H$ and $K$.
