Can I operate my own private tours inside galleries, such as London's National Gallery?

I would guess this would be a hard no at any museum or gallery. These organizations typically maintain complete control over how their material is displayed, presented, or viewed. Most that I've visited operate their own guided or unguided tours featuring a resident docent or recorded narration. You can always check with them to see if its allowed or to inquire about how to become a guide yourself, but expect them to refuse some random tour guide showing up and making money off of their collection. And don't try it without permission, as this will get you a quick exit, trespassing notice, etc.