Convert Logo to Print Files

I have only a photo of my current logo, so the first step is to recreate it as an exact-match vector. Once it is rebuilt, I need one small change: update the colour palette to the new tones I will provide... (Budget: £10 - £20 GBP, Jobs: Adobe Creative Cloud, Adobe Illustrator, Graphic Design, Illustrator, Logo Design, Photoshop, Print Design, Vector Design)