LONDON—U.K. Prime Minister Theresa May tried Monday to drum up support in Copenhagen and The Hague ahead of talks on Britain’s exit from the European Union, amid skepticism from the traditional allies as they grapple with anti-EU movements at home.

The trip marks her first meetings with EU counterparts since she said the U.K. would formally start the process of leaving the bloc by the end of March and signaled that curbing migration would be the priority. That stance has been met by tough words from European leaders who say...