Thanks to those who responded to my query. This is the answer that I received from the Immigration Office and may be of further help to those who have the same question.
Dear Mr. +++++
Thank you for your enquiry.
If an endorsement in your passport says Indefinite Leave to Enter, you are no longer under immigration control and are not required to apply for any further visas.
The valid 'to and from' dates on the visa is the time period permitted for you to enter in to the UK to activate the visa.
Upon entering into the UK the word 'enter' status changes into 'remain' by the activation of the visa.
If you have a new passport and require the Home Office to transfer your Indefinite Leave to Remain status into that passport you will need to obtain and submit a NTLTOC (No Time Limit) application form.
Application forms are available by calling 0870 241 0645 or from our website on
www.bia.homeoffice.gov.uk. All details and instructions are included with the form.
Yours sincerely
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The website states that it is not necessary to transfer the Settlement visa to your new passport but you are required to carry both if you choose not to spend the £160 in order to do so.