You don't mention where your partner is right now. (in terms of the itinerary) Is there some way to get into Haneda from where they are? Rather than the currently booked Narita option? Or skipping Tokyo entirely, since it appears to merely be a layover.

There's not much point "putting your foot down" because it's ultimately JAL's responsibility to get the passenger to London and EC261 won't apply. BAEC will make a good faith effort to reroute IME, but a prompt refund is their only legal responsibility if another airline cancels. (BAEC is effectively a travel agent here)