It's generally not possible to open a business account in the UK remotely. It's even difficult (near impossible) for a non-resident (even if a citizen) to open a business or personal bank account while visiting the UK.
A recent report says that it may be possible to open an account via Barclays Offshore in the Isle of Man. This requires a large deposit, and probably lots of paperwork and fees (most offshore locations have stricter "know your customer" rules than major countries). Note that while the Isle of Man is inside the UK banking system (for sort codes, account numbers), it is a separate territory that doesn't have the same deposit guarantees as the UK.
There is no legal reason why a UK company has to bank within the UK banking system, although many companies paying the company would expect it or require it, and an account in anything other than sterling would complicate the accounts. It could have an account in your home country.
It's not even a legal requirement that the company has an account in its own name at all. Some people use a (separate) personal account for this purpose. There are plenty of reasons why this is a bad idea (for example it's unclear who/what owns the money in the account, and can give the appearance of director's loans), but it's a work-around. Most inbound electronic payments only require a sort code and account number, the account owner name is not checked.
The UK does have a much simpler and cheaper company registry than most European countries, but the near-impossibility of opening a bank account for a business in the UK as a non-resident has made it an unsuitable place to register a small international company.