I would make a new reservation for the flight you want, cancel the old reservation and just check-in with the ticket as is. There is no requirement for the date on the ticket to match the reservation - the ticket is not endorsed "VALID DATE/FLT SHOWN". BA check-in might take the ticket to a ticketing counter to get a sticker put on, which always strikes me as rather unnecessary, but certainly if you are checking in at an F counter in LHR they will do this for you and it's no big deal.

I don't recall BA ever charging me GBP25 for a date change - they have charged me that in addition to the USD75 for a routing change when I didn't have time to get to another airline for them to do it. (But I may have forgotten - I've done a lot of these tickets - so if I did report it here then feel free to remind me.)