I would go for Durham, all the way.

The law school is ranked by the league tables as up there with Oxford, Cambridge, LSE, etc.

The law taught there is English, as opposed to Scottish. Scottish law is weird, and, frankly, a niche activity.

Also, you might be disappointed with the social life at Edinburgh - it does have something of a 'rah' reputation. The Brits there could be quite snotty.

Durham's a really nice town. If I wasn't going to LSE, I would consider it.

(All above is a personal view.)

Well, I'm not sure that's true. The best unis, with the best law faculties, in the UK are Cambridge and Oxford, probably by some distance. UCL and LSE are strong. (Imperial is great, but that's a science only uni.) Close behind, you've got Edinburgh, Warwick, Manchester and Bristol. Beyond that, there are a whole bunch of very respectable unis, including Durham, with good quality LLMs, and I'm sure if you went to one of those you'd enjoy it very much. What we're seeing today at many unis is greater specialisation in LLMs (e.g. LLM in International Law, as opposed to a general LLM), and a consideration then would be which is the best LLM of that type (which might not be at the 'top' law faculty, but which probably is at a faculty that has many leading academics working in that area).

When you do an LLM, you typically choose about 3 topics to study over the year; how much Scots law would be involved would depend heavily on the topics you chose; for example, if you chose International Law, there would be none, most probably. And probably little or none in IP, medical, Internet, Human Rights, and so on. Law Schools all over the world are pretty international, and want students from all over, so tend to offer courses of relevance internationally, not just for their jurisdiction. To be quite honest, I think you should choose somewhere you want to go, and where you'll be happy, rather than agonise over which is 'the better'; a future employer is likely to be far more interested in your LLM course grades and references from the academics who taught you, rather than whether your LLM is from Durham or Bristol or wherever - and you'll most likely get better grades if you're studying somewhere you're happy, with course modules you're interested in, than somewhere you think you 'should' be.

i would go for edinburgh though i am a wee bit biased since i attend the llm in commercial law in the lovely edinburgh.
in my opinion e. is the perfect mixture of excellent repudiated university and stunning city. commercial law offers a broad range of courses you can hardly find anywhere else in the uk (except the london law schools which share courses).
Compared to e. durham is rather rural but the college thing is nice i guess. regarding the credit of the law schools i think the international law firms favour edinburgh - this was what a german partner of freshfields told me. he said on their list are for universities, oxbridge, lse and edinburgh. law firms like linklaters, freshfields, gleiss lutz (germany) invite llm students from e. to london to get in touch. at these presentations i have only met students from oxbridge, london and edinburgh so these supposed to be their preferences.
some words about e. - the town is really amazing, (guardian and observer voted it in five consecutive years the best city of uk) either the landscape, with the sea and arthurs seat, an extincted vulcano. on the other hand the urban aspects are even more impressive. the city centre is divided into old town and new town, both world cultural heritage. old town is dominated by the massive castle which is a must see and the various small closes and winds - sidestreets to royal mile, which connect the castle and hollyrood palace (summer residence of the queen). in the evenings you find several ghost- and historic tours there.
the law school is located in the in the city centre so you can do anything by foot. other amenities are national gallery, botanic gardens, national museum of scotland, the harbour with lots of fish restaurants (but rather pricy) and the uncountable pubs in old town. george street, located in new town, provides also several fancy bars and night clubs (tiger lilly, le monde, shanghai - google it!). you should also try to play some holes of golf, which is a scottish obsession, and subscribe in one of the over onehundred student societies of the university of e. e. also benefits from the airport which grants quick access to many destinations around the world.