fine... how do I fix this...?

The correct answer would be that the garment is defective and should be returned to the store from which you bought it, in exchange for another whose dye is more permanent. If that is not possible, then you might consider trying to dye it, but the results may or may not be good.

What is the jacket made of? What kind of fiber is the embroidery done in? You cannot know what dye to select if you do not know what it is made of. A dye must be carefully matched to the fiber content of whatever you are trying to dye.

If the jacket is made of cotton, you can dye it with a cool water fiber reactive dye, such as Procion MX or Drimarene K dye, at room temperature with soda ash; if the stitching is made of nylon or another synthetic thread, it will not take the dye and thus will not be ruined. You could try Dylon Washing Machine Dye, since you are in the UK, using water of about 80° Fahrenheit (27°C). If the fabric and the stitching are made of the same fiber, however, they will both be dyed.

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