LONDON—High-level talks between the U.S. and the U.K. on strengthening trade ties will begin immediately, Downing Street said Saturday, following British Prime Minister Theresa May’s meeting with President Donald Trump in Washington on Friday.

Mrs. May’s office said a team of U.S. and U.K. officials would start scoping out what can be achieved together before the U.K. exits the European Union. Turkish President Recep Tayyip Erdogan, who Mrs. May met in Ankara on Saturday, made a similar commitment to increase trade links with...