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Thanks for your patience and enquiry. You are correct in that it is a simple procedure for the bank to notify the bank that the mortgage has been paid in full and as such the charge is to be removed. Practically the bank could also give you a letter/document to say the loan has been paid in full which could be accepted by the land registry as proof that they no longer hold a charge over your property.
I am not sure why they have said to your solicitor that they are unable to speak to him/her. All that is required is for you to give written consent to the solicitor to act on your behalf. Ask, your solicitor therefore whether they would like written authorisation to act on your behalf.
The only practical course I could recommend you taking is to write to the bank by way of a formal written complaint with respect to their reluctance and/or delay to act upon this request. Once a formal complaint is raised the bank is mandated to act. If they fail to respond to your enquiry or the resolution is unsatisfactory you may escalate matters to the financial ombudsman for it to review your complaint. In my experience, once a formal written complaint is raised with the bank they tend to be more proactive in their approach.
Can I clarify anything for you? I hope I have answered your query in a way that is simple and easy to understand. If anything remains unclear, I will be more than happy to clarify it for you. In the meantime, thank you once again for using our services.
You are welcome.