It looks like the UT program allows students to get their LLM at the law school at either the University of Nottingham or the University of Edinburgh. Both are very, very good schools and are well renowned in the UK (maybe not as well regarded as Oxford or Cambridge, but in my opinion they probably should be).
Note that Scots law is an entirely different system from English law, so unless you have your sights set on working in Scotland in the future, you'll want to choose Nottingham over Edinburgh.
Now, if you think you might potentially want to work in London in the future, there are two routes that will likely get you there. The first is taking a position whereby you'll practise US (most likely New York) law, but be based in the UK. The big firms in London do a lot of cross-border stuff, and as a result, most have a good number of US lawyers based in their offices. In order to take that route, you won't need to be qualified in England, so the LLM from the UK won't officially be of use. However, I'd think that it would be looked upon favorably because you'd have some link to the UK and a better understanding of the legal system that the firm is based in.
Hope this makes sense and is of some use.
The second route is for you to become an English lawyer. To do this, you'll first have to be licensed in one of 16 US states. You'll then have to take a test under the Qualified Lawyers' Transfer Scheme:
http://www.sra.org.uk/solicitors/qlts/apply.page. To do this, again, the UK LLM will be of no official use. But I would think it would be practically useful in that the test would be easier for you, you might be exempt from some parts of the test (check this), and you would be able to use the LLM to your advantage in your job search.
In sum, whichever route you choose to take, getting an LLM from the UK will not give you any 'official' advantage. At the same time, I would say it would be very useful in practice in either case.
The question this leaves for you is how the experience would be viewed to US employers versus spending an extra year at UT. My guess would be that it would probably be viewed neutrally most places. If you wanted to work in a global firm, it might be seen as an advantage.