I think you mean with Lax-Friedrichs the local Lax-Friedrichs or Rusanov scheme [1], [2] where the wave speeds (left & right-going) are given by the maximum eigenvalue $\lambda_\max := \max_i | \lambda_i(\boldsymbol{U}) | \in \sigma\Big( \boldsymbol{F}'(\boldsymbol{U}) \Big)$ where $\sigma$ denotes the spectrum of the Jacobian $\boldsymbol{F}'(\boldsymbol{U})$.
Note that the Rusanov / Local Lax Friedrich scheme is an approximate Riemann solver: For a conservation/balance law with Riemannian initial data, it provides an approximate solution. In particular, the scheme depends only on the trace (face / edge) values at the cell boundary / interface. In fact, it does not care how you got them - i.e., the scheme is agnostic whether you used simply the cell average value, a linear approximation with limiters or in your case the fifth order WENO scheme to reconstruct the face / trace values.
To finally answer your question, you need to take only the trace values $\boldsymbol{U}^-, \boldsymbol{U}^+$ into account at a certain face.
Then,
$$\alpha := \max \bigg\{ \max_i \big \vert \lambda_i (\boldsymbol{U}^-) \big \vert \in \sigma\Big( \boldsymbol{F}'(\boldsymbol{U}) \Big) \Big \vert_{\boldsymbol{U}^-} , \max_i \big \vert \lambda_i (\boldsymbol{U}^+) \big \vert \in \sigma\Big( \boldsymbol{F}'(\boldsymbol{U}) \Big) \Big \vert_{\boldsymbol{U}^+} \bigg\}. $$
In other words at each face $i$ you jest need the trace values $\boldsymbol{U}_i^+, \boldsymbol{U}_i^-$. In that case, your numerical flux (I think you lack the factor if $1/2$ in your formulation and confused the signs) is given by
\begin{align}
\hat {\boldsymbol{f}}_{i+1/2}& = \frac{1}{2} \Big( \hat {\boldsymbol{f}}_{i+1/2}^+ + \hat {\boldsymbol{f}}_{i+1/2}^- \Big) =\frac{1}{2} \Big( F(\boldsymbol{U}_i^+) - \alpha \boldsymbol{U}_i^+ + F(\boldsymbol{U}_i^-) + \alpha \boldsymbol{U}_i^- \Big) \\
&= \frac{1}{2} \Big( F(\boldsymbol{U}_i^+) + F(\boldsymbol{U}_i^-) + \alpha (\boldsymbol{U}_i^- - \boldsymbol{U}_i^+ \Big) \end{align}