Opening a bank account can get tricky here since many banks want to two forms of identification and only allow one of them to be from your home country, such as a passport. The other form typically has to be something showing a UK address, like a bill or (ironically) a bank statement, which many expats don't have in their first few weeks.
However, there are a couple banks in London which don't have that requirement:
Of course, once you do have a UK address there are many other banking choices available, including Halifax, Santander, HSBC, Royal Bank of Scotland, Standard Chartered, and others. The UK also has something called Building Societies. This is a member-owned bank that is similar to the credit unions of the United States. Each member gets voting rights and can attend and speak at meetings. The largest of these is Nationwide, which has branches all over London.
For now, savings deposits of up to £75,000 held in a bank or building society are insured by the European Union. This may change as the UK negotiates its exit from the EU.
There is no fee to withdraw money from any bank's ATM in the UK when using a debit card, regardless of whether it's your bank or not. Credit cards may incur a charge, however, when used for cash withdrawals at an ATM.
Helpful Money Resources
Tips for expats on foreign currency exchange.