In the U.S. we've had a couple of cases similar to this. Again, the UK's laws on the issue are different and may be interpreted by the courts differently, so these cases have no real bearing on this particular issue, but they do have bearing on the general discussion being held about whether a creative business can be made to create product they do not personally agree with.
Elane Photography, LLC v. Vanessa Willock
Vanessa Willock attempted to hire Elane Photography (a sole-proprietorship) to photograph her commitment ceremony. Elane Photography refused on the grounds that she was unwilling to produce expression (photography) that expressed something she disagreed with. The Human Rights Commission found this to be incompatible with New Mexico's public accommodation laws, as it was de facto discrimination against Vanessa Willock based on her sexual orientation. The verdict was upheld by the New Mexico Court of Appeals. The SCOTUS refused to hear further appeal.
GLSO vs. Hands On Originals
I think this is a better analogue, but it hasn't gone the distance, yet. The Kentucky Gay and Lesbian Service Organization attempts to hire Hands On Originals (a t-shirt shop) to produce t-shirts for the Lexington Pride event. Hands On Originals refuses because they don't agree with the message and don't feel they should have to print messages they don't believe in (eg: should not be required to express speech they do not agree with). The county Human Rights Commission has found the refusal of service to be a violation of Kentucky public accommodation laws. Further litigation is pending.
There are a couple of other cases percolating in the same neighborhood, including atheists suing a t-shirt shop, and a gay bar suing a print shop. Frankly, most of these get handled before they get to real court. Either the business goes bust, or the people discriminated against just move on, or a local HRC tells them to quit being dicks and they get the message.
These cases are part of what's driving the desire for laws that allow people to discriminate against customers based on religious reasons, because while it's not entirely settled law in the U.S., the trends have been pretty clear if you follow the cases. Regardless of what excuse proprietors are using in any particular argument, it usually comes down to: Is the business refusing service -- and for the purposes of the courts this means any service that the business extends to any patron -- to people of a protected class. If so, the proprietor usually loses.
On this particular case, which is going to be determined by an entirely different set of laws, the local Human Rights Commission has already weighed in, so I suspect the next step will be litigation, and we'll just have to wait and see how it turns out.