if administering TTM properly, the only stages of the ich life cycle that will be present are not possible to interfere with the TTM regime.
Stages are:
Trophont / when attached to the fish / 3 to 7 days attached to the fish
Protomont / crawls on substrate / 2 to 18 hours
Tomont / begins to encyst on a hard surface / hardens in 8-12 hours
Tomites / while a cyst, it splits into daughter parasites / 3 to 72 days inside cyst, however most take 4 to 8 days
Theronts / new parasites hatched from the cyst / 24 hours to find a fish host or die
TTM exposes the Tomite stage. This stage cannot transfer via any water transferred given this stage is concreted to a surface somewhere.
Regarding Theronts -- you won't see any of these since the whole point of TTM is to prevent Theronts from ever emerging.
If you happened to transfer Protomonts or Tomites over with the water, it does not matter since they will still require the time it takes to form a cyst/Tomite and hatch/emerge, which takes well over 72 hours... would have transferred by then.
So.... looooong story short, there is in theory no need to wash your hands for Ich purposes. But as said above, never hurts.