Nancy Gribble and Nero Wolfe (at least the TV version played by Maury Chaykin) both use British pronunciations like "tomahto" and "shedule" rather than "tomayto" and "skedule", and yet both seem to be Americans (although Nero Wolfe originated from Montenegro). How unusual is it for Americans to use "British" pronunciations, and what kind of reaction does this get?
As a native speaker of American English who has been exposed to a variety of media, it is generally unheard of for an American to say "tomahto" and "shedule" rather than "tomayto" and "skedule". The only time I've ever come across "tomahto" pronunciation with any regularity is when people say something such as "Tomayto, tomahto", generally implying that we agree on an issue even though we may have a different approach. A tomato is a tomato no matter how you pronounce it.
It is possible that there are certain registers in the US which use the British pronunciations, but I haven't run into any American born persons in the last four decades which do so, even after extensive travel around my country and in varying social situations with people from all walks of life.
As far as the reaction of Americans to British pronunciations, I don't believe this part of the question is definitively answerable without conducting research which accounts for register, education, and class. I tried various Google searches to attempt to find such research, to no avail. I can only tell you my own reaction.
If an American uses the British pronunciations of "tomato" or "schedule", the reaction is ultimately dependent upon the situation and how well I know the person. It might just be a goof, or that person may think that using the British pronunciations somehow make them sound more intelligent. This only applies to native speakers. I expect non-native speakers from countries such as India to use the British pronunciations because that is what they were taught in school.
I may be a good person to answer your question. My mother is British, but I grew up in New York. For the first 5 years of my life I spoke with a British accent, used British words (zed, lorry, telly). After that my accent became neutral-american, but I still use lots of British words and pronunciations, including 'tomahto'. (I don't believe I've ever said 'tomayto'.)
When I say 'tomato' most people ask me to repeat it, which I do. Many many people ask me why I don't say it properly, so I explain. Many people assume I do it as a way to be eccentric, which I don't. When I'm around the family I find myself slipping into a more British vocabulary and pronunciation. I use both 'scedule' and 'shedule', depending. I'll call someone a 'chap' or a 'guy' or a 'bloke' or a 'dude'. I'll say 'bollocks' or 'crap' about equally.
So, I guess the answer to your question is that many people assume it's an affectation. That's about the worst of it.