not sure about a few of the points regarding mobile app and API,
But if you are willing to either store £50k average in the account OR transfer £5k in each month (and you can transfer it back out) while holding two direct debits on the account you can get the following account without a monthly fee (£25 pm)
Citigold
http://www.citibank.co.uk/personal/banking/citigold/benefits.htm
Allows you to do sepa transfers for free and even gives you a 8 CHAPS transfers a month. All the while ofcourse allowing faster payment, you can have a linked EURO and USD account (http://www.citibank.co.uk/personal/banking/citigold/usdeuros.htm)
Alternatively hsbc do many international accounts... you may run into the fact you will potentially have lots of btc related payment in and out and issue with any uk bank though...