Masters programs in politics are usually quite receptive to law graduates (I did one after my LLB).
However, I have a feeling it would be difficult to get into an economics master's, since they seem to require a great deal of mathematics/statistics as a prerequisite for entry at the post-grad level. It is possible, at LSE I think, to do a foundation year before commencing an econ masters, in order to make up for a non-econ background. But I presume that's really only going to be worthwhile if you are dead set on a career as an economist of some sort.
A word of warning though - I remember a barrister telling me that his chambers might be a little suspicious of an applicant for pupillage with a non-law master's, since it might reflect less than a complete commitment to the law. As it happens, I don't actually agree (but then I don't make pupillage decisions!). Some chambers are indeed very diverse in the intellectual makeup of their members and, furthermore, in some cases I would have thought it a definite virtue to have a non-law, though still relevant, post-graduate qualification (e.g. an MA in Human Rights if you wanted to practise in that area, etc.).