Yep they will be british I saw a job application for a job with the Italian embassy a week or so ago and you could not be a dual citizen.

From what I have seen of the British embassy in Phil the mangers and ECO's are all Brits and the plebs locals.

Same in the Uk for foreign embassies

I had need about 5 years ago to send an email to the ECO due to changes which happened and due to the ECO it was directed to being off the manger replied and responded to emails i sent.

Dont forget as with all British cvill servants the will have KPI's and hopefully one of them is replying to enquires with in a certain amount of time.