We have a registered trade mark, does this protect our interior designer’s show home designs and mood boards that we present, and the installed show home in terms of the finished look?

The Trade Mark will cover you if someone uses that Brand to undertake the same work as you (always assuming the trade mark is registered in the correct classes). What it won’t cover is how something looks. That is Design and Copyright law. So the Trade Mark covers what something is branded as, Copyright and Design what something looks like.