Hi - not sure when you're travelling but tourist card from London Cuban Embassy (at High Holborn) is available in person from 0930 - 1230 Mon-Fri. Is it possible you can do that prior to departing LHR for MEX? You need proof of air ticket, accommodation and photocopy of your passport main page.
I am off to HAV next month but with IB ex PAR/MAD - I've sorted tourist card from Cuban Embassy.
Good luck