Do I need to fill in the tax form if I'm from the UK?

Sure, you will need to fill in the tax form, so that Adobe can apply the correct tax to your earnings. If you are taxed in the US or in the UK, that depends on the terms of the tax agreements between the US and the UK. I would guess, you would be taxed in the UK.

As taxing is complex, you should for further questions consult a professional in that matter.