Maybe not in the UK (although there probably is), but there certainly is a trademark for Sherlock Holmes (the character) in the U.S. See record 78461204 at
www.uspto.gov, click on trademarks, then go to TESS and search for "Sherlock Holmes". You will see that the character
is covered by trademark in the U.S.:
"IC 009. US 021 023 026 036 038. G & S: photographic, cinematographic, optical, measuring, life-saving and teaching apparatus and instruments, namely, magnifying glasses and disposable cameras; calculating machines, namely, digital calculators; computers; computer hardware and computer software,namely, computer game software;
recorded digital versatile disks featuring material concerning and relating to the literary characters Sherlock Holmes, spectacles, glasses, sunglasses, monocles, blank and recorded video cassettes,
audio cassettes and compact discs featuring motion picture films and audio books concerning and related to the literary character Sherlock Holmes and video game machines for use with televisions"
"IC 016. US 002 005 022 023 029 037 038 050. G & S: paper, cardboard, and goods made from these materials, namely,note pads, note books, posters, postcards, greeting cards; printed matter, namely,
books concerning and related to the literary character Sherlock Holmes, film stills and photographic prints;photographs; stationery, pens, namely, fountain pens, ball point pens, pencils, namely, propelling pencils, book marks"
The trademark is owned by The Sherlock Holmes Memorabilia Company Limited CORPORATION UNITED KINGDOM Palladium House, 1-4 Argyll Street London UNITED KINGDOM W1F 7LD
As far as copyright is concerned, if the book is in the public domain in the UK, it is in the public domain in the U.S. And vice-versa, if it is NOT in the public domain in the U.S., it is also NOT in the public domain in the UK (with a few very narrow exceptions).