is a Mann Whitney test on data where assumptions aren't satisfied as or almost powerful as a t-test on data where assumptions are satisfied?
A phrase like 'as powerful' doesn't really work as a general statement.
Power isn't especially comparable across different distributional models. The size of a given effect has different meanings in different parts of the distribution. Imagine you have a distribution that is pretty peaked, but has a heavy tail; by what measure to we say a particular size of deviation is similar to something with a much 'flatter' centre and smaller tail? A small deviation might be about as easy to pick up, but a large deviation might be (relative to the other distributional possibility we're trying to compare power for) harder.
With two possible sets of normal distributions, one pair with a large s.d. and one with a small s.d., it's easy to say 'well, power will just scale with standard deviation; if we define our effect size in terms of number of standard deviations, we can relate the two power curves'.
But now with differently shaped distributions, there's no obvious scale choice. We must make some choices about how to compare them. What choices we make will determine how they "compare".
For example, how do I compare power when the data are Cauchy with power when the data are say a scaled beta(2,2)? What is a comparable effect size? The Cauchy below has more of its distribution between -1 and 1 and less of its distribution between -3 and 3 than the other one. Their interquartile ranges are different, for example. What is our basis for comparison?
If you can resolve that conundrum, now consider if one of the distributions is skewed left and the other is bimodal, or any of a myriad number of other possibilities.
You can still compute power under any particular set of assumptions, but comparison of one test across different distributional assumptions rather than two tests under a given distributional assumption is conceptually very tricky.