The fluorescent tube as we know it today has in fact been around for a very long time indeed.  A number of experiments relevant to the invention of the fluorescent tube as we know it today have been undertaken dating back to the very dawn of electric lighting at the end of the 19th century.  

It was really in the 1920s however that more intensive research into the commercial introduction of such a lamp started as all of the necessary components were there.  Cheaply produced glass tubing, phosphors which emit visible light when excited by shortwave UV, proven electrode designs, and the means to create the necessary control gear.  While it's hard to pin down a precise date without doing further research - which one day I probably will - it appears that it was somewhere in the early 1930s that mainstream production of fluorescent lamps as we recognise them today would have started.

Since then there have been inumerable improvements to the electrode, ballast and phosphor designs which have improved both the overall system efficacy and the colour rendering abilties of the lamps, the overall design hasn't changed much however - if you were to look at a tube from the 40s, you would still most recognise it as a fluorescent tube.

Philips TL 8W/35 with Pink Coated Tube

Unknown Manufacturer - F8T5 Blue
Iwasaki Eye G8T5 Germicidal Lamp