Your comments are correct for a single ticket. When it is 2 separate tickets, the first airline is not obligated to verify the validity of the 2nd ticket and can treat the pax as O/D and not in transit at the destination ... requiring all of the appropriate documentation for entry into the destination airport, LHR in your example. Never mind that the pax swears they are in transit. This is the downside to having separate tickets. The airlines often treat them as one ticket, as a courtesy, but it isn't a right or contractually enforcable. Most of the time it doesn't matter, but your example illustrates the danger of using separate tickets. I've never had visa problems, but I have run into missed connections and subsequent problems with the 2nd ticket that were my problem rather than the problem of the airline that caused the delay. So it isn't an academic consideration, these problems happen some percentage of the time (admittedly very low).