Perhaps there is another Atlantic gulf here - but I am not sure that I am wholly clear about "corny" - the usage you suggest indicates to me something different from the way I would use "corny".

To me "corny" implies a fashion that has become dated. An old joke is "corny". Those metal bars with a little weight on the end that held shirt collars down are now corny. A song expressing old fashioned popular values may be corny.

"Sappy" I don't know of being used in England in the way you imply. "Sappy" to me would indicate young flexible wood growing with vigour, and by analogy a young person's vigorous pushing aside of the old.

"Twee", to me means excessively socially sweet or "nice" (but not "nice" in a good way), cloying, saccharine - overly inoffensive, self-consciously winsome. Beatrix Potter's books and animals in Victorian dresses are twee. The Rev. W. Aldry's books (the original "Thomas the Tank Engine") are twee. "My Little Pony" is twee. Most things chintzy are twee. Constance Spry verses in greetings cards are twee. Cards for mothering Sunday (a specified date on the Christian religious calendar) calling it "mother's day, or worse, "Mummy's day" are twee. A woman of 30 or over calling her mother "mummy" or her father "daddy" is twee. A man of over 30 doing so is nauseating. I would not equate twee with daintiness or delicacy, and certainly not with quaintness which implies old-fashioned.

"Cute" to mean "attractive" I also find nauseating. I object to its colonisation of the English language. It can mean acute or adept (as in "a cute trick"). It is a word I almost never use, and certainly never in the American way which I also find imprecise (maybe because I am too busy vomiting to think about its meaning). The use of the word "cute" however, is twee. Where others might say of a baby girl that she is "cute" I would say "pretty" or "attractive" or "fetching".

I hope that clarifies what I was meaning.