You may struggle with the mathematics requirements of an elite finance MSc program unless you take more classes beforehand, but you could also go for something softer like 'finance and X' (where X=business/investment/banking/management/etc) instead, its not like many private sector organisations are likely to know or care about the difference.
And yeah as interrobang says, you dont need a finance MSc (or any finance background really) to get into a finance/banking related grad scheme. It does help to have a prestigious university on your CV though.