Personally, out of the two, I would go for Leicester over Essex.
Not only is the reputation better, you will be better off as more law firms tend to visit Leicester as opposed to Essex. This would provide you with greater networking opportunities with law firms.
I can't say much about the city itself, but I'm sure others can provide some input on that.
I do know that Leicester is a pretty diverse city. I'm not too sure about the area around the University of Essex.
As for study abroad exchange programs, here are the universities that Leicester offer for law:
http://www2.le.ac.uk/offices/international/overseas-exchange/outgoing/departments-contacts/law.
And here are the universities that Essex offer for law:
Australia - Curtin University of Technology
Australia - Macquarie University
Australia - Monash University
Australia - Queensland University of Technology
Australia - University of Western Australia
Austria - British Council Assistantship - Austria
Austria - Universität Wien
Belgium - British Council Assistantship - Belgium
Canada - British Council Assistantship - Canada
Canada - Brock University
Canada - Queen's University
Canada - University of Lethbridge
Canada - University of Regina
Canada - University of Saskatchewan
Canada - Université Laval
Chile - British Council Assistantship - Latin America
China - British Council Assistantship - China
Christchurch Polytechnic Institute of Technology
Colombia - Universidad de Los Andes
Cyprus - European University Cyprus
Czech Republic - Univerzita Karlova v Praze
France - British Council Assistantship - France
France - Université Jean Moulin Lyon 3
France - Université Paris Ouest Nanterre La Défense
France - Université Toulouse 1 Capitole
Germany - British Council Assistantship - Germany
Germany - Freie Universität Berlin
Italy - British Council Assistantship - Italy
Italy - Libera Università Internazionale degli Studi Sociali (LUISS) 'Guido Carli'
Netherlands - Erasmus Universiteit Rotterdam
Poland - Uniwersytet Warszawski
Russian Federation - National Research University Higher School of Economics
Spain - British Council Assistantship - Spain
Spain - Universidad Autónoma de Madrid
Sweden - Stockholms universitet
Switzerland - British Council Assistantship - Switzerland
Turkey - Koc University
United States - Bloomsburg University of Pennsylvania
United States - Brooklyn Law School
United States - Indiana University of Pennsylvania
United States - Louisiana State University Paul M. Hebert Law Center
United States - Washington College of Law, American University