I'd use Illustrator's 3D effect for a base.
Draw a series of rectangles.....
Select them all and drag them to the Symbols Panel
Then draw a circle, just a standard circle, and choose
Effect > 3D > Extrude and Rotate. (I filled my circle with yellow so it can be seen.)
Click the Map Art button, navigate to the correct side, and then pick your symbol of rectangles. Then click Scale to fit and check the Invisible Geometry option. . .
OK twice to apply the effect.
Object > Expand Appearance from the menu, then choose
Object > Expand. You may need to also
Object > Ungroup.
Ultimately this gives you a series of consistently placed curves which can then be altered with different fills or other manipulations to create the overall type.
The benefit of using 3D here is that you can rotate and adjust the 3D object to create unique curve angles without much trouble. There's obviously some further hand manipulation in you sample. However, this should provide a solid, consistent, base for adding cross bars and end caps manually.
If you want more of a "wrap" to the curves, play with rotating and scaling the Map Artwork in the 3D effect. It takes some adjustment to get things to align properly, but it can be worth it.