I think the most likely answer is BA and LHR. I find it easier to believe that, all aspects of this considered, that they would have allowed the wife to go airside for an hour on the honor system there than at SVO. But that's just a guess.

Not really to help the OP, but just as a thought exercise, I wanted to talk more about what someone said about the issue of that she should have been caught before departure from VCE. I'll use LHR/UK as the presumed transit location just for the sake of discussion, even though it may have been somewhere else.

Someone mentioned that the airline probably violated their duty to the UK, by transporting a TWOV passenger there that lacked the proper documentation for admittance to their next destination, thus making them ineligible for TWOV (and based on nationality, inadmissible to the UK). If she didn't already have a ticket from LHR back to SIN, and didn't have the means to buy one, what would they have done with her? First of all, could have and would have the UK fined the airline for transporting the passenger to LHR in this case?

Separate from that, what would they have done with the passenger? Based on whatever treaties or other arrangements cover such things, could the UK have sent the passenger back to VCE (possibly at the airline's expense, given that they should not have transported her)? Would Italy have the ability and obligation to "undo" her exit, since she was legal there immediately before departure, and just restore her back to legal status as if that flight hadn't taken place? Or, would the UK only be able to send her to a place where she could be lawfully admitted?