Having somebody else pick up your bag is a bit of a no-no from a security PoV, so I don't think HND will allow a short-check-in of your bag to LHR if your connection to EDI is very short. You could ask for it anyway, knowing full well you won't pick up the bag and that your partner will, and the handler in HND my allow it, but it might cause a bit of a problem for your partner if she were to be stopped by customs at LHR, or if the bag were to go missing. Plus, obviously, there's the problem of her having to check-in three bags at LCY, with the extra cost that it might entail.