Hi filthiefist- I'm not sure why you objected to the first answer from bednobs. It wasn't hearsay- it was true statement about what can happen if you have only short service. Of course, if you have been there several years then this won't apply in your case, but as you haven't said what your length of service is we don't know whether this applies in your case.
The follow up answers from Bednobs are also helpful.
Relevant factors in whether the dismissal would be deemed fair or unfair woold be whether the employee had made similar mistakes before, whether he had been warned, what similar action had been taken with him/her and others previously, what the consequences were of the late arrival, what facilities were available for double checking rotas, etc
Buildersmate, Barmaid, Ethel and Buenchico may help unless your last reponse to bednobs put them off.