Forget letters, you will automatically get assessed by HR and graded low down - at least for the big companies.
Find a friend, or friends, or friends of friends, or friends of relatives (etc...) already in companies and find out if they have an internal referral process, if so get them to put you forward with a great reference, it may at least get you to the first interview, after that a lot is how you perform in that interview.
Also - a lot of companies place more emphasis on the theory that A level results correlate more closely to future potential than degree results, so if you have good A level results highlight them in your communications
Find out from Exeter if they have recruiting links or research links with any consulting companies, if so use those links to get yourself seen by recruiters and / or partners there.
If none of that works, target companies desperate to find analysts with a financial background - often companies with sold work but not enough people may compromise on their hurdle standards if they like you, and will be willing to give you a trial period to prove your worth. And finally, try temporary employment agencies who place people on projects - not a stable career but will build experience.