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2020 Statistics/Mathematics PhD Profile Evaluation

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Undergrad Institution: Imperial College London (Mathematics 4 year MSci integrated masters, graduated June 2018)
GPA: ~3.9 to 4.0 equivalent
Type of Student: International Male
Masters Institution: Oxbridge (Masters in Mathematics, currently attending, will apply to schools next year with grades from this in hand)
GRE General Test:
V: 167
W: 4.5
GRE Subject Test in Mathematics:
M: 92%
Programs Applying: PhD in Mathematics and Statistics 
Research Experience: Two summers with different professors in the stats department, one of which led to a paper being published. Expository final year thesis in probability.
Awards/Honors/Recognitions: Deans list.
Pertinent Activities or Jobs: Internship as a quantitative trader. Will be working as a quantitative researcher analyst from July 2019 (hopefully only for a year then entering grad school).
Letters of Recommendation: 1st from the stats professor where the paper was published, 2nd from an analysis professor (supervisor of thesis), 3rd from an algebraic geometer. Should be strong.
Relevant Grades: I thought I'd give a description as I will be an international applicant. I didn't officially take the courses written in italics (its rare if not impossible to be given permission to take extra courses for credit, certainly not for this many) but my letter writers verify that I studied them to a high standard independently either under themselves or their colleagues. "A" grade equivalent in everything but a couple courses taken in the first year. 
Analysis -- Analysis I, Real Analysis, Complex Analysis, Measure and Integration.   (Functional Analysis, Analytic Methods in PDEs, Fourier Series and Theory of Distributions, Stochastic Calculus)
Algebra -- Algebra I, II, III, IV sequence (starts from undergrad vector spaces/linear algebra, groups and rings and builds to homological algebra and graduate groups/rings). Then Galois Theory, Lie Algebras, Commutative Algebra.   (Infinite Groups, Group Representation Theory, Modular Representation Theory)
Geometry/Topology -- Metric Spaces and Topology, Algebraic Topology, Algebraic Geometry, Differential Topology.  (Manifolds, Riemannian Geometry, Complex Manifolds)
Probability/Statistics -- Probability and Statistics I, II, (measure theoretic) Probability and Markov Processes. Then Applied Probability, Time Series, Generalised Linear Models.   (Statistical Theory, Statistical Inference)
Applied stuffs -- Methods I, II, DEs, Multivariable Calc/Fourier/PDEs, Applied Analysis.   (Function Spaces and Applications, Advanced Topics in PDEs)
I have not yet decided what courses I will take to examinations this summer, but it will likely be a mix of analysis, stats and geometry right now.
Applying to where: Only really looking at top programs in either field tbh. Especially given that I may go back into industry at some point, branding is pretty important, and supposing I do want to come back to the UK it'd be very useful to have an internationally renowned name behind me.
Statistics - CMU, Berkeley, Stanford, Harvard, Chicago, Columbia
Mathematics - Berkeley, Stanford, Harvard, Columbia, Chicago, Princeton, MIT.
I'm not really sure if my profile is suited more to statistics or maths programs honestly - while I have research in statistics, and my job is also statistics heavy, my coursework and extra reading is dominated by maths.
Does anyone know what the calibre of incoming students at these programs is roughly like? Having checked the syllabi and done some timed past papers I know I could pass Harvard/Stanford quals and Berkeley prelims right now, and judging by the level of depth in some transcripts I'm almost ready to take Berkeley/Princeton style oral quals with concentration in say representation theory and algebraic geometry. But my concern is, if given the plethora of talent that these schools can select from, is basically everyone in my position or better with regards to mathematical knowledge upon entering the program? Because if so, I'm quite worried about my lack of original research now.
Thanks if you read, any thoughts would be greatly appreciated!
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Given your exceptional performance at ICL and Oxbridge (which both rank among the top 10 schools in the world) and the fact that you were able to publish a paper with a statistics professor, I anticipate you will have no difficulty getting into a top program in Mathematics or Statistics in the U.S.A. I think you are actually in the top tier of the students most likely to be admitted to these programs: having already taken a handful of classes that would typically be taught only at the PhD level in the U.S. *AND* having published a paper in your BS/MS program. The vast majority of PhD applicants cannot claim both of these feats, so your profile is very impressive. 

For the top PhD programs in Statistics in the U.S., I think you are in good shape. Past mathematics coursework counts for a lot more than statistics coursework in PhD admissions for Statistics (similarly the case for PhD programs in fields like Economics and Finance). I think the rationale is that someone with a strong math background can learn the Statistics/Econ/Finance part at the PhD level no problem, but this may not be true of someone who has a degree in one of those fields but who has a light mathematics background.

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You have about as good of a profile as any statistics applicant will have, so you'll get into a few, maybe even most or all of those statistics programs. It is impossible to understate how much more difficult it is to get into a top 5 math PhD, though.  I think not have math research will make it tough to get into a school like Princeton - you should go ask mathgre forum for more input. You are in amazing shape for stats programs though. 

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