Who do you bank with currently?
They are probably your best bet, as you have an existing relationship with them - yes, you may have been living in your OD which isn't great, but hopefully you have a history of remaining within it (ie. not bouncing payments, exceeding your limit etc.). Yes they will likely look at your external credit reports too, but their internal scoring will carry a lot of weight.
That said, do you need a specific student account? If you are in a strong financial position, and expect to have a surplus as mentioned above, then it seems unlikely you would need the use of the interest free OD facility - so perhaps your existing account will cover what you need?