It looks like there are typically two earlier flights EDI-LHR before the one that arrives at 10:15. Perhaps the sensible thing to do is get the agent to change to an earlier connection to give them more time to transfer between terminals. IMHO even if there is a small cost involved, it may reduce the stress associated with worrying about whether they will make it.

PS You might also want to look at the return flights too, so they have a smooth end of holiday.