The answer to that question is a maybe, while some banks -- RBS, Nationwide, Santander, HSBC and Barclays -- do allow customers to change their contact address and landline/mobile telephone number via their online website, there are still many that require customers to either visit them at their branch or telephone them to make such a change: such as Lloyds, TSB and Birmingham Midshires (for their savings accounts). Due to the threat of ID theft and fraud, to change a contact address and landline/mobile telephone online will typically require the customer to enter an additional piece of information -- beyond simple logging in -- such as: entering a memorable word or an answer to a question (they created when registering the account) or by entering a unique code from a card reader (such as Barclays PINsentry).
An example of a bank which does not allow customers to update their personal details on the Internet is TSB.
(Pictured: published on TSB's website on 30/10/18 at www.tsb.co.uk/help-and-support/general/)
Alongside visiting a branch or calling them, TSB also provide a PDA 'change to address' form that can be printed off, filled in and signed by a customer; to change details via the post. Most banks provide a 'change to address' form but they do state how important it is to sign the document; the Halifax and Lloyds state: "Please don't forget to sign the form as we will not be able to carry out your request without a signature. We use the signature to verify your identity against signatures we hold for you - if we can't match these, we will need to write to you." (Source: https://www.lloydsbank.com/help-guidance/customer-support/change-your-personal-details.asp and https://www.halifax.co.uk/helpcentre/manage-your-details/change-your-address/).
The following banks -- who do not allow customer to change personal details via the Internet -- provide a 'change to address' form and a telephone number (to change contact details) at the following webpages:
Bank of Scotland: https://www.bankofscotland.co.uk/helpcentre/manage-your-details/change-your-address/
In conclusion, changing personal details with a bank, such as an address or telephone number, can be easy but it can also be a frustrating and potentially scary process for some bank customers. Changing home is typically a stressful event in most people's lives, so having to provide a certified copy of ID and bills etc, can be time consuming and difficult when in the process of moving home -- especially if a person has been in temporary accommodation before the move and does not have the correct documents that the bank demands. Doubly frustrating when compared to banks who allow customers to change their address in minutes online. Its a trade off, as while it maybe inconvenient to change address/telephone numbers via a branch with ID, it may increase the security of the account; the banks have a difficult task striking the right balance.