Hi Leo, the problem with London and to a lesser extent Edinburgh is that there is such a vast range of restaurants and cuisines that we really need more information to help.
Are you looking for fine dining that will come to almost £200 a head including wine. Or classic British food? Or some of the ethnic cuisines that are so well-served here.
If you can give us where you're staying, the rough budget per head and an idea of the kind of restaurant you're looking for then we can hopefully point you in the right direction.
Please be aware there is a search feature in the upper right-hand corner of this page. You can do searches for both cities there and find many, many helpful suggestions already here for you to review.
You will also find a number of relevant past threads listed directly below. It might also be helpful if you said what sort of price range is suitable for you.
If you like seafood, J. Sheekey is very good and not far from Trafalgar Square, but it is on the expensive side. The Great Queen Street pub is good and moderately priced.
Here is a link to an article in the Financial Times reviewing Great Queen Street and 3 other London restaurants.
Here is a recent thread that should be of interest: