In America the first flavored, non alcoholic beverages were popularized as a direct response to prohibition. A decent number of brewers took to making "Near Beer" or something of the sort (they never actually called it "near beer" because they each had their own marketing name for the beverage) which was essentially beer without the alcohol created by either boiling off the alcohol off the finished beer, or freezing the beer and pouring the unfrozen alcohol off (This is how most modern non-alcoholic beer is made). These beverages gained a modest level of commercial success but given how easy it was to obtain alcohol during prohibition, there really was no long-term staying power.

In terms of something closer to a soda, or soft drink we also look to prohibition. Coca Cola really attempted to ramp up production during prohibition looking to fill the void of beer, even going so far as to advertise mothers feeding their children Coke or other products. Coke, of course, existed before prohibition (interestingly, also as a alcohol-alternative. But if you were to try to promote a beverage before the mid 20th century it would have to be as an alternative to alcohol) but it was during prohibition that they really tried to corner a market.

Source: Last Call: The Rise and Fall of Prohibition by Daniel Okrent

I would argue that shrubs could count as... well, not technically carbonated, but the effect is very much the same. Shrubs are a drink comprised of flavored vinegar (or just good tasting vinegar, like apple cider vinegar in the case of switchel), sweetening, and water. [edit: or rum/other alcohol.] It tingles on the tongue just like carbonation. Vinegar drinks are very old, and I do not have information about a start date.

Recipes exist from the period of the Victorian temperance movement for drinks that mix acidic ingredients with basic ones to make it effervesce.

Miss Beecher's Domestic Receipt Book, 1850: Effervescing Fruit Drink Very fine drinks for summer are prepared by putting strawberries, raspberries, or blackberries into good vinegar and then straining it off, and adding a new supply of fruit till enough flavor is secured, as directed in Strawberry Vinegar. Keep the vinegar bottled, and in hot weather use it thus. Dissolve half a teaspoonful or less of saleratus, or soda in a tumbler, very little water till the lumps are all out. Then fill the tumbler two-thirds full of water, and then add the fruit vinegar. If several are to drink, put the soda, or saleratus into the pitcher, and then put the fruit vinegar into each tumbler, and pour the alkali water from the pitcher into each tumbler, as each person is all ready to drink, as delay spoils it.*

Effervescing Jelly Drinks When jams or jellies are too old to be good for table use, mix them with good vinegar, and then use them with soda, or saleratus, as directed above.*

Things A Lady Would Like to Know, 1876:

Raspberry, Strawberry, Currant or Orange Effervescing Draughts.-- Take 1 quart of the juice of either of the above fruits; filter it, and boil it into a syrup with 1 lb. of powdered loaf sugar. To this add 1 1/2 oz. of tartaric acid. When cold, put it into a bottle, and keep it well corked. When required for use, fill a half-pint tumbler three parts full of water, and add 2 table-spoonfuls of the syrup. Then stir in briskly a small tea-spoonful of carbonate of soda, and a very delicious drink will be formed. The colour may be improved by adding a very small portion of cochineal to the syrup at the time of boiling.

The first two taste horrible, the last one is very nice. Shrubs in general are very nice, actually. The Victorians loved a raspberry shrub, but my favorite is apple cider vinegar flavored with lemon peel, simple syrup or honey, and ice water. It is like sparkling lemonade.