Peel the lemons with a small knife, trying to get as little of the white pith as possible. Put the lemon peels in a large serving bowl and add the sugar and muddle. Let the lemons and sugar sit for about 1 hour.
When the hour is up muddle the peels and the sugar again. Add the lemon juice and stir until the sugar is dissolved. Stir in the port, rum and cognac and mix to combine. Add the water and a large block of ice (you can freeze water in a smaller bowl the night before or just use available cubes). Grate some nutmeg over the top and ladle into sherry glasses.