Just graduated from Engineering school - want to move into Finance, thinking of doing MSF.

Hey guys, I just recently graduated with a first class honours degree (70%) from a non target UK university ( think Queen Mary London) in Mechanical Engineering. I checked the GPA converted and a first is equivalent to a 4.0GPA. As I've been progressing throughout my degree I found myself more interested towards finance and I want to pursue a MSF from a target University.

The downside is that I have no relevant work experience, I undertook a 1 year internship with Procter and Gamble, but it was in Product Supply and completely irrelevant. In addition to that, I have a Gap year now until I can hopefully start my MSF. I have started studying my GRE and am aiming to achieve 165+ in Quant and 160+ in Verbal.

I am also applying for off cycle internships in any relevant finance field, but I am not sure of my odds of getting any. It's difficult for me cause I need visa sponsorship to work in the UK (USA passport). I've applying to firms in the US but no response so far, I think it's cause I have a UK degree.

I'm hoping to get into a target school in the UK or USA ( Oxford, LSE, MIT, Columbia, Stern...). I aim to work in the buy side ( Hedge fund ) after my degree.

Do I have any good chance to get into the schools I'm aiming for? What else could I do during my Gap year if I can't land an internship? Studying for the CFA could be better than nothing for me, but landing a relevant internship would significantly strengthen my application.

Any advice would be appreciated.