Masters in mathematics after bachelors in mechanical engineering
Solution 1
This sort of dilemma is often country and culturedependent. Some areas of the world will be far more strict about what is / isn't a 'maths degree'. Usually here (UK) I see it phrased as more like 'a degree with a high quantitative component' which includes computer science, physics, engineering, etc, but I don't know much about the Indian academic culture.
My recommendation would be to just contact someone at an institute you're interested in and ask them. Either they say yes, and now you know, they say no, and you don't need to bother applying, or they don't answer, in which case the only thing you lost was the time sending the email. This also might vary by instutite, it's unlikely any answer will apply to every institute, even within the same country.
Solution 2
I think your plans for your future are quite reasonable, but my thoughts count for very little. Many engineers have a mathematical bent of mind and I believe many mathematics departments (especially in applied fields like statistics) will recognize that. An example of an engineer who became a mathematician is: Raoul Bott. In all probability, you are not going to be Raoul Bott (prove me wrong), but I think it's worth pursuing the career goals you want. Your job, is to make a convincing case for yourself and to find the right department for you. You can do this by talking with people who have achieved what you are trying to achieve, getting their feedback on their department and your application (especially statement of purpose).
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Comments

Smokey 11 months
I am in a bit of dilemma regarding my career aspirations.
I am a mechanical engineering graduate from one of the top 5 IITs in India.
I have a average CGPA of 8.87/10 (top 20 in my class).
I have taken close to six mathematical courses like: Maths1(grade A), Maths2(grade B), Transform Calculus(Excellent grade), Partial Differential Equations( Excellent grade), Probability and statistics( grade A), Control systems( Excellent grade).
Currently I am working as a quantitative analyst for credit risk team at an investment bank, the job here has drawn me closer to the field of mathematics and statistics and through a lot of selfstudy, I have decided to pursue a masters course in statistics/applied mathematics and possibly use that masters degree for phd applications later.
I am in my midtwenties currently. I have self studied the entire statistical Inference book from Casella and Berger to supplement my knowledge/fill in the gaps.
However the problem lies in the application part for the masters program as they require an undergraduate degree from mathematics and letters of recommendation, while I can arrange for letters of recommendation from my manager here at workplace and a few professors whom I did my bachelors project with/took their courses, however I am not sure how relevant they will be.
I understand that because I don’t have a undergraduate in mathematics, even if I get admissions, they won’t be into any top coveted programs but I am willing to put in the extra effort/fill up the gaps using my experience in that college for graduate school admissions later on.
Does this sound like a plausible plan or am I working for a lost cause?
Do I stand a chance with my profile for admissions into decent masters courses in mathematics in US?
Can attending a decent mathematics masters program work as a stepping stone into some good graduate programs?
Any insights would be helpful. Thanks in advance.

Smokey about 3 yearsthanks for your reply and suggestions. I will get in touch with people on professional networking sites like LinkedIn as well as departments themselves for their feedback and suggestions as well.

E. Rei about 3 yearsThis is not correct. There is far more to your academic background than a test that's not used very much across the world. Far more likely is that very few will care about your GRE test.

Smokey about 3 yearsThanks @H.Green for your suggestion, I will try this method out for all colleges I am interested in UK, US and Canada as well.