Wash and dry the cucumber. Grate about 1/3 of the cucumber into a small bowl. Use a mandoline slicer or sharp knife to thinly slice the remaining cucumber.
Grate the onion into the same bowl as the cucumber. Stir the salt and lemon juice into the grated onion and cucumber. Set aside.
Put the whipped cream cheese into a large bowl.
Use a strainer to drain the grated cucumber and onion. Be sure to press as much liquid out as you can.
Add the drained cucumber and onion, the chopped dill and the Worcestershire sauce to the bowl of cream cheese.
Use a rubber spatula to mix the cucumber spread until well combined and smooth.
Cut the crusts of the bread. Cut each piece into four even triangles.
Spread a thin layer of the cucumber sauce on each piece of bread. Top half the bread slices with two or three halves of cucumber slices. Then top them with the other half of the bread slices.