Yeast washing is to separate the trub (break proteins, hop material and dead yeast) from the viable yeast. It does remove some of the hop flavor, but I will tell you for sure that a starter made from washed yeast will still have some hop aroma/flavor...bear in mind you're diluting this into 5 gallons (or more), so it's negligible...
The issue with re-using yeast with mead is the ABV. My understanding is that re-using yeast from a brew greater than 7-8% ABV runs the risk of introducing off flavors because of yeast mutation...this probably varies by strain though, and I would bet that some strains could probably be more safely and reliably be re-used than others, but I'm sure not able to tell you which ones.
Regardless, in an apocalypse situation, I'd reuse the yeast!