The APD may be waived on separate tickets if the are considered linked by the airline.
"The connected flights must be detailed on the same ticket or conjunction tickets to qualify for the exemption. Tickets can only be regarded as conjunction tickets if:
(a) they are in one booklet; or
(b) where they are in separate booklets: each refers to the other and states that they are to be read in conjunction or there is a summary of the flights constituting the passengers journey including the flights in question."
http://www.publications.parliament.u...70704h0011.htm
"... the concession is not restricted to passengers who book all their flights with one airline. Any airline can offer through-tickets, including for passengers who have connected flights with different airlines, if the airline collects enough information both to link the tickets and to account to Her Majesty’s Revenue and Customs for the connection between the tickets. Many airlines offer that service to their passengers"
Basically, if an airline is willing to say that 2 tix are linked and can provide some evidence if needed, then the tix can be considered linked. It would seem that's what happened in Lili's case, perhaps in an informal manner.
But then an airline may not even need to provide evidence. Accounting options for the APD allow airlines to estimate "connected flight exemptions" so that they don't need to account for each individual connection.
http://customs.hmrc.gov.uk/channelsP...HMCE_CL_000508
Given that both of SST's tix are issued by AA, it would be a simple matter for AA to say that the tix are linked. It doesn't hurt to ask. Same applies for Keith Flyer, with the AA/BA link.