Do I need an ESTA to travel to Mexico from the UK?
No, it is only necessary if you are flying (or transiting) the United States.
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In addition, foreign visitors who, regardless of their nationality, have a valid permanent resident card from Canada, Chile, Colombia, Japan, Peru, the United States of America, the United Kingdom or any of the countries comprising the Schengen Area, do not require a Mexican visa.
Visa processing takes approximately 21 workdays from the date on which the application is received by the Embassy. A further 2-3 workdays should be allowed for the return of your passport.