I would think you wouldn't warp the logo in the file you send to print. The cup will more than likely be printed via a flexographic printer where their prepress department will warp it if necessary. You definitely should talk to them about it.
This doesn't mean you would not warp the artwork onto your existing template to show how the cup would look like when printed. Two different files. I typically would write down the settings I used to create the template and keep it in the job jacket for future use such as this logo warp.
It all rather depends on how big the logo is relative to the cup.
If it is small no-one will notice if it's not warped. If it is big you will probably need to warp it.
Here's a suggestion if you want to warp all your artwork:
Measure the printing area shown by the dashed orange line (coloured Art Lines 2). The height is easy. The width should be measured along the curve in the middle – you can do that close enough using a ruler on a printout.
Draw your artwork (square, not warped) to fit these measurements. You may need to make a background rectangle (not necessarily coloured) depending on the artwork.
Copy the dashed orange line.
Copy/drag your (square) artwork and arrange it in the stack so that the copy of the dashed orange line is above it. Select both.
Now do Object, Envelope Distort, make with Top Object and the copy of your artwork should fit into the printing area.
If everything looks o.k. you can expand the warp envelope.
I note that the pdf opens as RGB colour. You should probably change it to CMYK as soon as you open it in Illlie.
Thanks so much for this it helps a lot
The logo I need to warp to the template is made up of a script style font and when I warp it by the method you showed me it almost doubles in size is there something different I should do for a text logo ?
Also if you have to place it off centered you draw a rectangle no fill or stroke that ill fill the template when distorted and place the logo to the relative position group that with the logo and apply the envelope distort.make with warp to that group.
Did you remember to make a background rectangle? The one I made was 240 x 109.811 mm. It doesn't need to be filled or stroked but should be there for the distort to work properly. The actual size doesn't matter much but the proportions should be right. Your logo should sit on the rectangle in the correct size and position for what you need. If it needs to be repeated (say twice) the repeat increment should be (half) of the width of the rectangle.
I'll try that thanks a lot!