I've been a member of faculty at two of the universities that you list, not in Law, but a cognate area.
You are right to be cautious about league tables, not just because positions change year by year, but because they attempt to give an objective measure to what is necessarily a subjective judgment. The "student experience" on the one hand and prospective employers' evaluation are both inherently subjective.
To be honest the teaching quality - which is the crucial issue for you as student - is much the same between these institutions. A key thing to look out for is what proportion of your classes will be taught by full faculty staff and what proportion by teaching assistants (often PhD students, not necessarily bad, often very good). In other words, if you choose a place to study because it has some big name academic lawyers you may find that they're not actually doing much teaching, and certainly not of undergraduates.
In terms of employability (in law) the key issue is not which of these universities that you went to but the quality of the degree that you graduate with. The prospect of a training contract with a law firm (and possible sponsorship for LPC) will be almost exclusively dependent on the class of degree it looks like you'll get. Until the last year or so, the big law firms swept up people from Oxford and Cambridge - with the financial crisis even these graduates are having to go to job centres.
If there are particular parts of law (eg international, employment, banking, human rights, asylum, criminal) you should check that there are a decent range of options available which reflect your interests.
The final thing that I'd say is that choosing between law departments/universities may ultimately rest on non-educational matters. For example, do you want to live in a campus university out of town (Warwick) or do you want to study in a big metropolitan centre (Manchester, Birmingham), or do you want to mix with Oxbridge wannabes (Durham, perhaps Bristol).
For what it is worth - and I don't think it is worth much, because as I said all these departments are of roughly the same standing - my casual selection would be :
University of Nottingham 1
University of Birmingham 5
University of Warwick 3
University of Durham 6
University of Manchester 4
University of Bristol 2
Having said all that, I would seriously question the wisdom of seeking guidance on university choice from an internet forum
