First of all, they’re both excellent universities and the reputation difference between the two isn’t big enough that it’ll make a difference to future employers.
What you should base your decision on is which one is right for you as they’re very different universities.
LSE has a smaller student body, doesn’t have a campus as such and you’re living in London. Some people will be suited to that, some less so.
Notts on the other hand has a massive student population, a huge (and beautiful) campus, while you’re studying in a fairly big city.
They’re going to be completely contrasting experiences - LSE wouldn’t be for me at all but for you it might be. Base the decision on where you’d rather see yourself studying.