There is a difference between "plagiarism" and "copyright infringement". Plagiarism means pretending that you created something when you didn't. If your graphics designer claims he created your logo, and in reality he paid some art student to create it, that would be plagiarism. Which would be legally totally fine. (Or you claiming that you created the logo and not your graphics designer; your designer might be unhappy about it but none of RHS's business. )
What you would need to worry about is copyright infringement or trademark infringement. Trademark infringement would happen if people look at your vehicle for example and think "look, there's an RHS vehicle". With "Goodwin Garden Services" on your van that's unlikely.
The other is copyright infringement; if RHS claimed that your graphics designer copied their logo. The colors are similar but not the same, and using green colours for a gardening business seems natural. Your leaves are one shape, the RHS leaves are all different shapes and not the one of your logo. RHS has apples in their logo. The stem is very different, and yours uses the I of GoodwIn for the stem. The one similarity is the leaves being sparse on a white background. All in all I wouldn't worry too much.