^I don't know how anyone can broaden the appeal of somewhere like the Lake District.
It is the natural beauty of the place which appeals.
You either like it or not.^
Lots can be done.I have a holiday cottage that I rent out in Windermere, so I know the area well. There are many visitors from China and Japan and other countries around the world, but very few ethnic minorities from the UK.
I think widening the ethnic mix starts with introducing people to the Lakes (and other National Parks) at a young age, as my DH and I were. I was lucky enough to also go with my family, but well remember some less fortunate class-mates just running and gambolling, amazed at so much open space. Schools need to open up these opportunities to all children. Some will like it and return many times, some will decide it's not for them.
Regarding disabled visitors, there are already some excellent examples. There is a viewpoint overlooking Windermere, Orrest Head, that has been made accessible to pushchairs and mobility scooters almost all the way to the top. There is currently a push to do that last little bit. At Tarn Hows (a stunning beauty spot), the National Trust has a mobility scooter hire scheme and you can get all around the tarn by scooter. In other places there are walks for visually impaired visitors.
It is much too complacent to say it's there, people will find it, there is always room for improvement!