Before this movie devolves into a sweet but somewhat generic rom-com, it has a pretty heavy opening in which Richard E. Grant's character deals with the death of his wife in childbirth and has three crying scenes.  Can't see a lot of tears, but the heavy emotion is there.  In real life, Grant had a daughter who was stillborn, so this can't have been an easy subject for him.  
In his memoir Grant talks about being offered "menthol" during filming of "Withnail and I."  He thought he was being offered cigarettes, then was told that it was for his eyes, to produce fake tears,  He declined, noting that crying on cue is easy for him.  He also talks about being an easy crier in real life. 
It's too bad he mostly does comedies.  
The scenes are at 14:04, 17:14 and 39:48.