I think it has much more to do with the number of people tested and the criteria they're using for doing the test. It's not fair to compare on this basis, just as it's not fair to compare nations on the number of confirmed cases.
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It would be an indication, but it's not struggling *today*. It's certainly not at capacity. It has about 3000 beds per million people, which is more than enough to cope with the serious subset of the 20 infections per million... *right now*
I think it's a bigger factor that a larger proportion of Britain's total cases have concluded (many in death) than Switzerland's. (If we assume that the known/unknown ratio is the same, which I'm not sure of.)