I think she may be right that there's nothing you can do.
I wouldn't give a flying fluff if my neighbour stuck a 20 feet long model of the Starship Enterprise on the roof, and painted a starfield on the back. And I certainly wouldn't care if someone painted their house blue.
However, you're not happy @Katey83 so you are free to report it. (To the council.) Be prepared for a long-running war with your neighbour though, as if you do manage to get the council to tell them to repaint it in a basic colour, they will hate you forever.
Here's what I found on Google.
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In the UK, you generally do not need planning permission to paint the exterior of your house, as this usually falls under Permitted Development. However, painting it a bright or unusual colour comes with a few major caveats depending on your local area and property type.
Key Rules and Restrictions
Listed Buildings: If your home is listed, you almost certainly need Listed Building Consent from the council for any exterior painting, even if you are just re-painting it.
Conservation Areas and Designated Land: If you live in a Conservation Area, an Area of Outstanding Natural Beauty (AONB), or a National Park, permitted development rights are often more restricted. In some of these areas, councils enforce an Article 4 Direction, which removes your automatic right to change the exterior colour without applying for planning permission.
Leasehold or Social Housing: If you are a tenant (e.g., in a council or housing association property) or own a leasehold, you must get written permission from your landlord or freeholder before altering the exterior.
Restrictive Covenants: Check your property deeds for any restrictive covenants that dictate acceptable exterior colours.
Next Steps & Local Checks
Before you pick up a paintbrush, it is highly recommended to double-check the specific rules for Eccleshall and your exact address.