From your description, it has elements of "sci-fi" but they don't sound like those elements don’t really drive the story.
A good definition of SF is that something is SF to the extent that the story is driven by fictionalized science (in the broadest sense of any rational, objective, experimental or observational body of knowledge) or technology. Thus a story isn't necessarily "SF" or "Not SF", but to some degree SF.
From your description there is an SF element, but it does no more than set up a situation which plays out according to the rules of a mundane 30's British comedy. So while it would seem to have a touch of SF about it, it's not very stfnal.
(Obviously if someone who has read the book has details that change my impression from the OP's description, my conclusions might well change.)
Added: I'm going to (considerably) expand my previous answer in response to comments:
Asking if a story either is or is not SF is not very helpful. Some stories not only have stfnl elements, but literally couldn’t be told without them. Others have some stfnl elements, but those elements are more part of the scenery than essential to the story. Another story might be a mainstream novel that happens to have, say, an impossible invention or mentions in passing that there is a moon base. (If any anachronism makes a story SF, the SF canon is a lot bigger than most people think! Shakespeare has clocks in Julius Caesar and billiards in Antony and Cleopatra!)
If the definition of being SF is a story having any SF elements at all, then there’s a lot that is SF, but I don’t think that definition is particularly helpful.
Even less helpful is an attempt to divide the world of the fantastic into two disjoint categories “SF” and “Fantasy”! Never mind that for the first fifty years of SF it was considered to be a subset of fantasy frequently called “science fantasy.” Even today when the definition of fantasy has evolved there are plenty of stories which are ambiguous. (For an old example, consider de Camp and Pratt’s “The Mathematics of Magic.” Is it SF? Is it fantasy? Both answers are clearly “yes”, but this is only possible if they are not disjoint sets.)
It’s easy to come up with any number of similar cases which further shred the idea that there are two categoris, one called “SF’ and one called “Fantasy”.
All of these problem cases go away if instead of asking if a story is SF, asking to what degree it is SF and to what degree is it fantasy. Some stories are very stfnal, some are just a little bit, some are nearly pure fantasies, while others – sometimes claimed for the mainstream as “Magic Realism” – spice up a basically mundane story with some fantastic elements. Some are basically mysteries which are also slightly stfnal. (Look at J. D. Robb’s very successful Mystery/Romance/SF In Death series!)
So I suggest that a story is SF to the extent that it depends on fictionalized science or engineering with both terms extended to include any rational, objective, experimental or observational body of knowledge. A story is fantasy to the extent that it depends on fictionalized magic or the supernatural with the terms again being inclusive. Something can be either, neither or both and can be just a bit fantastic or a whole lot fantastic.
A story like Hal Clement’s “Uncommon Sense” has science so deeply built into it that it is SF to a very high degree. But something that’s basically a Cowboys and Indians story where they ride hroten, and shoot blasters and are menaced by the Greenies, is hardly SF at all (Planet Stories was full of this sort of thing!)
So to properly answer the question, I think you need to ask two things, first is there any explanation for the body switching beyond “it just happened at the dentists’”. Secondly, are the consequences – either personal or societal – of the body switching explored, or is it just the setup for normal Wodehousean hijinks?