First and foremost, it's the consumer's responsibility to regulate the amount of sugar they consume from beverages. That's it, pure and simple. Despite the jumping up and down that Action On Sugar make, none of us are living in a Nanny State where governments don't trust us to regulate our sugar intake and we are free to decide for ourselves if we wish to drink these high sugar beverages. It's your call.
The sugar content figures of the products sold in coffee chains are readily available at the branch last time I looked and should there be any doubt, the staff at the chain will clarify the issue. If you don't know, ask. Again, it's your call.
The so-called sugar tax implementation is indeed aimed at manufacturers rather than coffee shops and food retailers. I should know as I was consulted on sugar metabolism and other issues some time before the tax became headline news. I can tell you that there are major potential problems in attempting to extend the tax to coffee shop drinks that you cite.
Where would we stop with this? Do you consider that a cafe should somehow pay the sugar tax because it provides sugar dispensers on the tables? How would the tax be implemented in this case? How far off would such action be from taxing the 1 Kg bags of sugar we come home with from the supermarket? Would you find it acceptable to deny your neighbour a second teaspoon of sugar in the coffee you've provided when he/she pops round based purely on the cost of that teaspoon of sugar?
You need to keep a sense of perspective over sugar consumption. It'll continue to be your call for some considerable time and rightly so as governments do consider that the average consumer is blessed with some degree of common sense.