Set a saucepan with 500ml water over a medium heat. Once hot, add the sugar and stir, without boiling, until it has dissolved. Remove from the heat and leave to cool.
Juice the cucumbers in a juicer or blitz well in a food processor. Pour into a sieve set over a bowl and press with the back of a spoon to strain as much juice as you can. (If you used a food processor, discard the pulp.)
Transfer the cucumber juice to a large jug. Add 300ml sugar syrup (or to taste) and the lemon juice, stirring to combine. Top up with 1 litre chilled sparkling or still water.
Pour into ice-filled glasses and garnish with the cucumber sticks and lemon slices.