First of all, Hello to everyone.
Secondly, my comment is purely my own opinion and in now way represents any official opinion or policy of any service provider.
Is it racist to request a white doctor? Yes, absolutely!
Is it racist to request a British doctor? Let us first of all throw out the window any notion of colour prejudice, because requesting a British doctor can in no way guarantee a white caucasian doctor! Even the word 'caucasian' refers to European descent, not British. Many, many British doctors are of ethnic-minority descent.
Is it xenophobic (which is considered racist) to request a British doctor? Yes, absolutely!
However, the confusion might arise because the problem is not correctly defined, or poorly communicated, and an inappropriate solution identified or demanded.
E.g. in Sodthisforfun's example, the problem was not defined, explained or communicated, and an inappropriate solution, potentially misconstrued motive, applied. Under these circumstances the service provider is totally within their rights to dismiss such a request.
(So it is not only speakers of foreign languages that have difficulty in making themselves properly understood.)
If the problem (of extreme difficulty in hearing) had been communicated and a request for any available assistance, I suspect the service provider (having several available options) would be happy to comply. As Securespark suggested, options are available, including a designated helper. I have intentionally avoided the term 'carer'.
Asking for a British doctor will not guarantee effective communication. Some foreign doctors (e.g. especially those from America, Australia, Canada, Ireland, South Africa etc,) may be more easily understood than a British doctor from say Ireland (NI), Liverpool, Scotland, Wales, Newcastle, Birmingham, etc.
Additionally, Scandinavian speakers of English frequently display little or no accent.
It is well known that speakers of all languages adopt the predominate accent of their teachers. For instance, children grow up speaking with the accent of their peers. Exactly the same applies to foreign students of English, they invariably adopt the accent of their teachers.
Only through contact with a wide variety of other English speakers do we lose our regional accents.
(Some retain the ability to seamlessly return to their regional accent, sometimes with intentional comical effect, e.g Lenny Henry.)
Is it reasonable to expect service providers to comply with requests for specific requirements?
How far should we responsibly go? Some hypothetical examples: my Mother has alzheimer's, should I expect (morally or reasonably) the service provider to accommodate her requirements. My son has a developmental disability, should I morally or reasonably expect the service provider to accommodate his needs?
Or should I morally and reasonably expect to provide some support to my relatives?
Finally, is it legal for an employer to allocate work based on the employee's nationality? Therefore, can a request to be treated by a British doctor be legally complied with? I suspect not.
God forbid that we are ever in any situation where a national identity determines the suitability of a person to carry out certain duties. That has all the connotations of nationalism, fascism, or nazism.