I've found a document by the BMA which seems to suggest that you would need to provide your consent for the private doctor to speak to your nhs one....

Are NHS GPs obliged to provide patient information to private practitioners?When patients self-refer to private practitioners, this is frequently followed by a request to the GP forany information that might be relevant to the treatment in question. The exchange of informationbetween those providing care for a patient, including liaison between NHS and private practitioners,is important.
• GPs’ primary concern should be for the interests and safety of their patients, with due regard toconfidentiality.
• Good communication between colleagues, with the patient’s consent, is required so that medicalinformation can be exchanged on the basis of a clear ‘need to know’ in connection with the care ofthe patient. This is in line with the GMC’s guidance, which states that ‘sharing information withother healthcare professionals is important for safe and effective patient care’.25
• NHS GPs should provide relevant information on request about the patient’s medical history orcurrent condition to other doctors providing care, including doctors working in the private sector.
• Patients should not be made to feel that they need to withhold information about private treatmentfrom their GP for fear of losing their entitlement to NHS care.
• If the GP is aware that treatment is being sought privately and has information that might affect thesafety or outcome of the treatment, this should be shared, with the patient’s consent.
• Failure to provide relevant information when the patient’s consent has been obtained could result ina complaint against the GP – either to the GMC or through the courts – if the patient is harmed asa result