tone of voice on the phone

Op
First of all, everybody has an accent, including you

What would you do if you have an emergency with your dog and your son is not available?

I would change vets to avoid this issue - what if your son does not have an answer to a question they ask? This liaison role he is in is very unreasonable

Different accents and immigration is a new reality that hit that hit the world many years if not decades ago - it is time to adapt

HiThere, there is such a thing as received pronunciation. While nobody's accent is perfect RP, some people are more able at speaking in received pronunciation, and can also speak in their local accent when appropriate . The received pronunciation is lingua franca.

Of course it would be better for me if I could adapt to foreign
and strongly regional English pronunciation but I never have been good at this, as a matter of fact.

It would be best for customers ands clients if receptionists were trained in received pronunciation.

As for emergency call- out to the vet, my vet has supplied a card from a local vet who specialises in emergency euthanasia where the dog lives so that the dying is as prompt and peaceful as can be