Barcelona for me!

If I may put a slightly different spin on the anniversary angle of the OP, I always find that time seems to pass more slowly in a country which doesn't speak as its first language my own first language. Which is nice if you only have one week.

You have to make a choice between two great cities. I would choose London though both are wonderful. If you are planning to spend a full week in one city, London is so full of things to do and see that it seems the best choice to me. While Barcelona is fun and exciting, we have never wanted to spend a full week there. On the other hand, we have spent a full week in London several times and have left without seeing and doing half the things we wanted to do.