For Education, in Guardian League Table for these 3 universities Birmingham is 19th, Sheffield is 26th and Sheffield is 65th, which is a big difference I think. The Complete University Guide puts Birmingham for 5th, Sheffield 28th and Leeds 32th.
Overall, I don't know how important are the league tables for you, but still a factor to consider. About the community and other stuff, I can't say anything because I'm also an international student and only applied to Leeds but for a totally different course.