I am currently pursuing a bachelor's degree in chemical engineering (2nd year student) from SLIET, Punjab, India, which is a Central Funded Technical Institute under the Ministry of Human Resource Development, India. Before moving on to the main part of the question, let me describe a bit about my background in mathematics.
So, I am currently involved in a group research project on discrete geometry, which is being mentored by a renowned mathematician (of UCSD) in the US. I will also be doing a research internship on algebra under a well-known faculty of IIT Bombay in the upcoming summer of 2021. I participated in PROMYS 2020 Bridge Program (as a junior counselor and researcher) and PROMYS Boston 2019 (as a first year student) as a Mehta fellow. Also, I participated in MSP 2018, ENS Paris, as one of the top 50 young mathematicians around the world. In addition to this I am a Regional Mathematics Olympiad (RMO) awardee and have also received a high-distinction in AMC 2016, back in high school.
Now moving on to the main part of the question, I wish to pursue my graduate studies in Mathematics from the US or Europe, that is, I wish to shift to mathematics after I am done with my bachelor's degree in chemical engineering.
But, as you might know the mathematics that is taught in a typical engineering program covers real analysis, multivariable calculus, Fourier series, linear algebra, vector calculus, complex analysis, differential equations (ODE and PDE), Laplace transform, probability and statistics and numerical analysis. We are not taught abstract algebra (groups, rings, fields, etc.), most part of analysis that is covered in a typical undergraduate mathematics course and topology. So, these are the courses that would formally lack from my coursework, which would seriously make my application (for graduate studies in mathematics) weak and less competitive.
Though I will be taking five additional undergraduate courses in mathematics and computer science next semester (from NPTEL), namely, introduction to rings and fields, introduction to algebraic topology, introduction to Galois theory, introduction to algebraic geometry and commutative algebra, and design and analysis of algorithms. I hope taking these courses would cover up for the lacking coursework to some extent. I plan to keep taking such additional undergraduate courses in mathematics and computer science for the further upcoming semesters too. In addition to this, I have also taken courses in Number Theory, Graph Theory and Topology during my two years at PROMYS.
I am also preparing for the GRE mathematics subject test and covering some of the undergraduate mathematics courses on my own, at the same time.
I have heard that funded masters are not a thing anymore in good US universities. So, should I be opting for an integrated PhD in mathematics from the US or should I be opting for a masters in Europe after my bachelor's? Which is easier to get into (provided my background in mathematics)? And, what else should I be doing (like more undergraduate research work, coursework, etc.) to enhance my profile in mathematics, in order to get into some decent universities?