There are no laws in the United Kingdom that prohibit financial institutions from working with foreigners. However, many banks prefer not to deal with non-residents because they have to follow Know Your Customer (KYC) procedures. Your application will most probably be refused in some financial institutions such as the Royal Bank of Scotland (RBS) without explanation, even if you have a long-term visa.
Other respectable banks, such as as Barclays, Lloyds and HSBC, are more flexible in working with foreign clients. However, irrespective of this flexibility, they will not overlook movements of funds on your bank account. When you get a large sum of money, they may check if the source is legal.