Here's some gov.uk (from https://www.gov.uk/government/publications/change-of-name-guidance/use-and-change-of-names) on the subject;
"Dual nationals
Normally dual nationals will be expected to align the name on their foreign passport or identity card with the names requested on Home Office issued documents.
However, where individuals can provide evidence that they are unable to change their name in their foreign issued documents, the Home Office will exceptionally consider allowing them to use a different name in a Home Office issued document (see Annex A). This may include married naturalised British citizens who are unable to change the family name on their foreign passport due to legal requirements. Such individuals may also be asked to provide evidence that they are living in the new name for all other official purposes.
Where, in exceptional circumstances, a different name is allowed for dual nationals then this will be recorded on Home Office systems and as an observation in the British Passport."