Which person(s) or legal entity/entities holds the U.S. and British
copyrights to Sax Rohmer's Fu Manchu stories and what is his/her/its
contact information?

Hello.

Copyright to Sax Rohmer's Fu Manchu is controlled by The Authors
League of America, Inc, in the United States, and by The Society of
Authors in Britain.

See this notation from a web page: 
"Fu Manchu © The Society of Authors and The Authors League of America, Inc."
http://www.pjfarmer.com/woldnewton/Links.htm

Two official trademark records at the U.S. Patent & Trademark Office
confirm that The Authors League of America, Inc., controls the U.S.
rights to Fu Manchu:

http://tarr.uspto.gov/servlet/tarr?regser=serial&entry=75602675
http://tarr.uspto.gov/servlet/tarr?regser=serial&entry=75344710

Here's the address listed with the U.S. Patent & Trademark Office:
THE AUTHORS LEAGUE OF AMERICA INC
330 W 42ND ST
NEW YORK NY 10036
United States

Here's a web page that lists the same address, plus a telephone number:

Authors League of America
330 West 42nd Street
New York, NY 10036
(212) 564-8350
http://www.musicals101.com/rightsearch.htm

I tested the (212)564-8350 number, and it's current.  The recording
indicated that you could leave a message for The Authors League of
America.

-------

As mentioned above, copyright in Britain is controlled by an
organization called The Society of Authors.

You'll see Sax Rohmer's name on the list of literary estates managed
by The Society of Authors:
http://www.societyofauthors.org/estates/

Contact information: 

The Society of Authors
84 Drayton Gardens
London SW10 9SB 
England

telephone 020 7373 6642 
fax 020 7373 5768 

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search strategy:
uspto.gov, fu manchu
"fu manchu" "Authors League "
"the Society of Authors", rohmer

I hope this helps.

Very prompt and full answer - everything I was looking for!