I'm not a member of the GW team, and I don't know.
One might say that the distinctive colour and typeface of the Gardener's World logo form a brand identification, and are some form of intellectual property. In your question, you recognise that because you refer to the image as a 'logo'.
But I can't understand what reason anyone would have for wanting to use the Gardeners' World logo on their website.
It wouldn't make sense to say, for example, that you are helping to advertise Gardeners' World. Far more people know about GW, and visit this site, than visit yours.
So why bother?
You seem to want to use it as a way of implying your credibility (as a gardener, I assume). This would be not far short of immoral and certainly deceitful, since you are using someone else's proven expertise, without their permission, to suggest that your practice is somehow approved by Gardeners' World. How would you explain this to a prospective customer?
There's no recommendation like a satisfied customer.
T.K.Dukes is a gardening business -courtesy of google