The short answer is yes, you can do it, since the TOST methodology is not restricted to t-tests. The p-value is the larger of the two p-values. A quick Google search led me to a methodological article (Meier U. Nonparametric equivalence testing with respect to the median difference. Pharm Stat. 2010 Apr-Jun;9(2):142-50) describing this procedure in detail.
First question: UMP is, nomen es omen, most powerful. If both the sample size and the equivalence region are small, it may happen to the TOST that confidence intervals will hardly ever fit into the equivalence region. This results in nearly zero power. Also, the TOST is generally conservative (even with an $1-2\alpha$ confidence interval). Whenever the UMP exists, it will always have power $> \alpha$.
Second question: Sometimes an UMP doesn't exist. It is this strictly total positivity of order 3 that has to hold for the density, see the appendix of Wellek's textbook on equivalence and noninferiority tests. Intuitively, this condition guarantees that the power curve of the respective point hypothesis test has exactly one maximum. Then the critical values are the points where this power curve has level $\alpha$. That's why you find them with this $F_{1,n-1,\psi^2}$-distribution in this question: Obtaining $p$-values for UMP $t$ tests for equivalence.
Also if your equivalence hypothesis is not standardized, i.e. $\mu \in ]-\epsilon, \epsilon[$ instead of $\mu \in ]-\frac{\epsilon}{\sigma}, \frac{\epsilon}{\sigma}[$, then even for normally distributed data an UMP has a strange rejection area in the $(\hat{\mu},\hat{\sigma}^2)$-space. See Brown, Hwang and Munk (1997) as an example.
The most important is, as you mentioned, that confidence intervals on the observed scale are more instructive than $p$-values. So the ICH guidelines require confidence intervals. This leads automatically to the TOST, because if you supply a confidence interval to the $p$-value of the UMP, confidence interval and $p$-value may contradict each other. The UMP may be significant but the confidence interval still touches the hypothesis space. This is not desired.
In conclusion, if you use the equivalence test "internally", i.e. not for direct scientific reporting but only as part of some data mining algorithm e.g., UMP may be preferable if it exists. Otherwise take the TOST.
Best Answer
The logic of TOST employed for Wald-type t and z test statistics (i.e. $\theta / s_{\theta}$ and $\theta / \sigma_{\theta}$, respectively) can be applied to the z approximations for nonparametric tests like the sign, sign rank, and rank sum tests. For simplicity I assume that equivalence is expressed symmetrically with a single term, but extending my answer to asymmetric equivalence terms is straightforward.
One issue that arises when doing this is that if one is accustomed to expressing the equivalence term (say, $\Delta$) in the same units as $\theta$, then the the equivalence term must be expressed in units of the particular sign, signed rank, or rank sum statistic, which is both abstruse, and dependent on N.
However, one can also express TOST equivalence terms in units of the test statistic itself. Consider that in TOST, if $z = \theta/\sigma_{\theta}$, then $z_{1} = (\Delta - \theta)/\sigma_{\theta}$, and $z_{2} = (\theta + \Delta)/\sigma_{\theta}$. If we let $\varepsilon = \Delta / \sigma_{\theta}$, then $z_{1} = \varepsilon - z$, and $z_{2} = z + \varepsilon$. (The statistics expressed here are both evaluated in the right tail: $p_{1} = \text{P}(Z > z_{1})$ and $p_{2} = \text{P}(Z > z_{2})$.) Using units of the z distribution to define the equivalence/relevance threshold may be preferable for non-parametric tests, since the alternative defines the threshold in units of signed-ranks or rank sums, which may be substantively meaningless to researchers and difficult to interpret.
If we recognize that (for symmetric equivalence intervals) it is not possible to reject any TOST null hypothesis when $\varepsilon \le z_{1-\alpha}$, then we might proceed to make decisions on appropriate size of the equivalence term accordingly. For example $\varepsilon = z_{1-\alpha} + 0.5$.
This approach has been implemented with options for continuity correction, etc. in the package tost for Stata (which now includes specific TOST implementations for the Shapiro-Wilk and Shapiro-Francia tests), which you can access by typing in Stata:Edit: Why the logic of TOST is sound, and equivalence test formations have been applied to omnibus tests, I have been persuaded that my solution was based on a deep misunderstanding of the approximate statistics for the Shapiro-Wilk and Shapiro-Francia tests