Yes, it also depends on the fare basis for your purchased economy-class ticket from LON to MIL. Only the most expensive tickets will be changeable right up to departure (and on some routes it seems only Club Europe class tickets are that flexible).
So you are effectively doing two changes:
(1) changing the underlying purchased economy-class ticket, for which there's usually a charge (if it's at all possible) except for the most expensive fare basis, and
(2) moving the miles-for-upgrade associated with the ticket, which you can't do at all within 24 hours of departure. (I don't know if there's a charge associated with doing so earlier - probably depends if you do it offline or online, in particular.)