First, I hope you carefully consider your options for this upcoming school year. If you have any more questions, please do not hesitate to ask me.
Now, here are my answers (much of it is from my past discussions with my former university's Graduate Vice Chair, i.e. he is in charge of Math PhD admissions for the university and he is a Full Professor; and what I have read online):
Top 15 in the UK (according to the QS rankings) would be equivalent to the Top 70 in the US (according to the QS rankings and US News rankings as there is some correlation between their rankings of US universities. The university you have been admitted to at the UK is decent/good. I can tell you that admissions in the US and UK can be sometimes be unpredictable (as my friend and I have both been rejected by a few lower ranked programs but accepted by some highly ranked programs).
By "different standards", I do mean that Math Subject GRE scores are generally higher for international students. That is why I mentioned that you need to have a high score/percentile on the actual exam (720+). Have you taken the practice exam provided online? The Math Subject GRE is a Tough exam (since it tricks students with the questions it gives and not that much time is given on the exam). For this reason, Math Departments tend to judge students by their Subject Test scores. Public universities tend to give priority to Americans (due to funding restrictions), so I would recommend adding additional Private universities to your safety list (in addition to the Public universities you would have on the list).
As long as you meet the minimum General GRE scores for Verbal and Writing for your desired Math Department (each Math Department sets a different minimum scores for Verbal and Writing). The Quant section is very elementary beginning Secondary School math, so with some (or a little) study, you should do very well.
The US News rankings are generally accurate in terms of grouping universities into various tiers (for instance, the difference between Princeton and Berkeley is marginal). These slightly recent rankings should also help (though they correlate with the top 50 rankings of US News) http://www.ams.org/profession/data/annu ... ey/group_i
The US News rankings and AMS groupings are reliable as they reflect the general tiers (of programs) and selectivity (usually correlated with program quality) of programs. Ranking is even more critical for students choosing to attend US Programs since most students have NOT chosen a single research area (like Topology, PDEs, or Analysis), yet.
Most US Math PhD Programs DO NOT interview you! Some PhD Programs, such as UNC Chapel Hill, will interview you if you are in their waitlist. Interviews for US PhD Programs (in my point of view) would not be useful because American undergrads and sometimes International students do not know which research area they would like to work on and they would not have much research experience (not even a thesis) in general.