From my perspective, it depends.
It depends on how much of what type slurry you're thinking of pitching. Are you going to stir up that last few millimeters of beer into the yeast cake to make the slurry? Are you going to cast onto the yeast cake? If so, you might note a bit of roasted-malt astringency.
If you're just going to stir up the yeast cake and last few mm of beer into a slurry and pitch a certain amount of that slurry into your fresh wort in a different fermenter, I wouldn't worry at all.
It also depends on how bright was your stout wort at cast/knockout. If there were lots of suspended solids in the wort - as though you made an extract beer with steeped specialty grains without a grain bag and a loose strainer, or had lots of hops particles in the wort - you'll no doubt have grain and/or hops particles in the yeast bed. Those particles might
impact the Mild wort, depending on the how much are in there and how they're distributed in the slurry you intend to pitch to the Mild.
Really, though, I wouldn't worry all that much about flavor; I should think you'd note more of a color impact than anything, and that only if you intend to cast onto the cake. If you're pitching a slurry, I wouldn't worry much at all.
But that's just me!
Edit: I've never washed yeast in this procedure; I've just either pulled it from the bottom port of a unitank (at work) or stirred up the cake into a slurry (at home, rarely) and pitched the result. Malkore, would you kindly explain your washing procedure? I'm curious.