Thanks for this excellent idea, Gealt.
Sweet Betsy From Pike with the refrain, sung at a moderate pace will get you about 20-22 seconds of hand washing and of course you can sing other verses for variety-

    Oh have you heard tell of Sweet Betsy from Pike,
    Who crossed the wide prairie with her lover Ike,
    Two yoke of oxen, a large yellow dog,
    A tall Shanghai rooster, and a one-spotted hog.
    Fol da dang did lee doo rall dee doo rall dee a.